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NUMERICAL MODELLING OF THE
UNDRAINED VERTICAL BEARING
CAPACITY OF SHALLOW FOUNDATIONS
by
VAN NGUYEN QUOC
A Thesis submitted for the Degree of
Master of Philosophy
at the University of Southern Queensland
October 2008
i
Certification of Dissertation
I certify that the ideas, results and analyses, conclusions reported in this dissertation are
entirely my own effort, except where otherwise acknowledged. I also certify that the work
is original and has not been previously submitted for any other award or higher degree to
any other University or Institute.
VAN NGUYEN QUOC, Candidate Date
Endorsement
Dr RICHARD SIMON MERIFIELD, Supervisor Date
ii
ABSTRACT
The bearing capacity of foundations is a fundamental problem in geotechnical engineering.
For all structures placed on a soil foundation, geotechnical engineers must ensure that the
soil has sufficient load carrying capacity so that the foundation does not collapse or become
unstable under any conceivable loading. The ultimate bearing capacity is the magnitude
of bearing pressure at which the supporting ground is expected to fail in shear, i.e. a collapse
will take place.
During the last fifty years various researchers have proposed approximate techniques to
estimate the short term undrained bearing capacity of foundations. The majority of existing
theories are not entirely rigorous and contain many underlying assumptions. As a
consequence, current design practices include a great deal of empiricism. Throughout
recent decades, there has also been a dramatic expansion in numerical techniques and
analyses, however, very few rigorous numerical analyses have been performed to
determine the ultimate bearing capacity of undrained soils.
In this study, finite element analysis has been used to analyse a range of bearing capacity
problems in undrained soil. The numerical models account for a range of variables
including footing size, shape, embedment depth, soil layering and undrained bearing
capacity of footings on slopes.
By using the powerful ability of computers a comprehensive set of solutions have been
obtained therefore reducing the uncertainties apparent in previous solutions.
iii
ACKNOWLEDGMENTS
I gratefully acknowledge the Research Scholarship from Vietnamese Government and
officers of The Committee on Overseas Training Projects, Ministry of Education and
Training of Vietnam.
I would like to express my deepest appreciation to Dr Richard Simon Merifield, Prof Thanh
Tran−Cong, Dr Amar Khennane and Assoc Prof David Buttsworth for their effective
guidance. Without their continuing dedication and encouragement this dissertation would
not have been possible.
Finally, I am indebted to my family, my parents, my wife Huong and my son Viet for much
support and encouragement over the years of my studies.
iv
PREFACE
The research work presented in this thesis was conducted in the Faculty of Engineering
and Surveying at the University of Southern Queensland from August 2004 to August
2006. This work was performed under the supervision of Dr Richard Simon Merifield.
During the term of the candidature, the following paper was published:
1. Merifield R. S. and Nguyen V. Q. (2006). “Two and three−dimensional bearing capacity
solutions for footings on two−layered clays.” Geomechanics and Geoengineering, Taylor
and Francis, 1(2) , pp.151−162.
v
NOTATION
All variables used in this thesis are defined as they are introduced into the text. For
convenience, frequently used variables are described below. The general convention
adopted is that vector and matrix variables are shown in bold print while scalar variables
are shown in italic.
A area of footing.
B footing width.
C1,� C2 constant coefficients.
c soil cohesion.
c� drained soil cohesion.
cu, su undrained soil cohesion.
c1, c�1 horizontal shear strengths at the top of the first and second layer
respectively;
c2, c�2 vertical shear strengths at the top of the first and second layer respectively;
cu1,� Ct undrained soil cohesion of top layer.
cu2,� Cb undrained soil cohesion of bottom layer.
c0 cohesive strength on the surface
D problem dimensionality, diameter of circular footing, the embedded depth
of footing.
Df footing depth.
De distance from footing to the crest of slope.
d�b ratio between thickness of the top layer and the width of footing.
E total number of elements in finite element mesh, Youngs Modulus.
Eu undrained Youngs modulus.
Fcs, Fqs, F�s, sc, ss shape factors.
vi
Fcd, Fqd, F�d, dc depth factors.
Fci, Fqi, F�i inclination correction factors.
F, FS, FR dimensionless factor of footing/soil interaction, that for smooth, rough case.
g, gi vector/components of prescribed body force, gravity.
H slope height, top layer thickness.
K relative shear strengths between horizontal and vertical direction.
L length of rectangular footing, distance from the edge of slope to footing.
Nc, Nq, N�, Nc��, Nc�, Ncq, N�q bearing capacity factors.
N total number of nodes in finite element mesh, stability number.
Nf stability number.
Nms, Nmc, Nmr modified bearing capacity factors of strip, circular and rectangular footing.
Ns slope stability factor.
N*c modified bearing capacity factor.
n relative shear strengths between the top of first layer and the top of the
second layer.
Qn net footing capacity.
Qu ultimate footing capacity.
qu, qf, qult ultimate bearing capacity/pressure of the footing.
q, qi vector/components of optimisable surface traction.
qnetbL�su ratio of net bearing capacity to undrained shear strength.
q uniform pressure, surcharge.
R radius of circular footing.
s1, s2 characteristic lines.
uj, u.j displacement/velocity components.
vii
urj, ur.
j rotational displacement/velocity components.
uif, ur.
if displacement/velocity components of footing.
�f soil shear strength.
� parameter of hyperbolic approximation of Mohr−Coulomb yield criterion.
� unit weight of soil.
� footing displacement, soil footing interface roughness/friction.
� change in soil cohesion with depth dcu�dz.
�cs empirical shape factor.
ij� stress tensor.
i� principal stresses.
� internal friction angle of soil.
�� drained friction angle of soil.
�u undrained friction angle of soil.
� Poissons ratio.
1contents
ABSTRACT ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ACKNOWLEDGMENTS iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PREFACE iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NOTATION v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. INTRODUCTION 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 BACKGROUND 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 BEARING CAPACITY OF FOUNDATIONS 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 THESIS OUTLINE 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. HISTORICAL REVIEW 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 INTRODUCTION 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 PREVIOUS EXPERIMENTAL INVESTIGATIONS 12 . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Undrained bearing capacity of surface footings 12 . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Undrained bearing capacity of embedded footings 18 . . . . . . . . . . . . . . . . . . . . .
2.2.3 Undrained bearing capacity of footings near slopes 20 . . . . . . . . . . . . . . . . . . . .
2.3 PREVIOUS THEORETICAL ANALYSES 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Undrained bearing capacity of surface footings 23 . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Undrained bearing capacity of embedded footings 30 . . . . . . . . . . . . . . . . . . . . .
2.3.3 Undrained bearing capacity of footings near slopes 38 . . . . . . . . . . . . . . . . . . . .
2.4 SUMMARY 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. UNDRAINED BEARING CAPACITY OF SURFACE FOOTINGS ONLAYERED SOILS 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 INTRODUCTION 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 PROBLEM DEFINITION 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 FINITE ELEMENT MODELLING DETAILS 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 ABAQUS FINITE ELEMENT ANALYSIS (FEA) RESULTS 51 . . . . . . . . . . . . . . . . . .
3.4.1 Footing on homogeneous clay 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Footing on layered clay 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 2−D Strip footing on layered clay 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 3−D square and circular footings on layered clay 62 . . . . . . . . . . . . . . . . . . . . . .
3.4.5 Effect of footing roughness 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 CONCLUSION 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS 71 . . .
4.1 INTRODUCTION 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 PROBLEM DEFINITION 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2contents
4.3 FINITE ELEMENT MODELLING DETAILS 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 ABAQUS FINITE ELEMENT ANALYSIS RESULTS 81 . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Shape factors 102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Bearing capacity and depth factor for strip footings 105 . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Bearing capacity and depth factor for square footings 107 . . . . . . . . . . . . . . . . . .
4.4.3 Bearing capacity and depth factor for circular footings 108 . . . . . . . . . . . . . . . . . . .
4.4.4 Bearing capacity and depth factor for rectangular footings 109 . . . . . . . . . . . . . . . . .
4.5 CONCLUSIONS 129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. UNDRAINED BEARING CAPACITY OF FOOTINGS NEAR SLOPES 131 .
5.1 INTRODUCTION 132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 PROBLEM DEFINITION 132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 FINITE ELEMENT MODELLING DETAILS 135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 ABAQUS FINITE ELEMENT ANALYSIS RESULTS 138 . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Effect of ratio L/B 166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Effect of the footing roughness 167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Effect of slope angle 167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.4 Comparison of this study with Narita and Yamaguchi (1990) & Kusakabe (1981) 168
5.5 CONCLUSION 168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. CONCLUDING REMARKS 170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 INTRODUCTION 171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 UNDRAINED BEARING CAPACITY OF SURFACE FOOTINGS ON LAYERED SOILS 171
6.3 UNDRAINED BEARING CAPACITY OF EMBEDDED FOOTINGS 172 . . . . . . . . . . . .
6.4 UNDRAINED BEARING CAPACITY OF FOOTINGS NEAR SLOPES 172 . . . . . . . . .
REFERENCES 174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1Chapter 1
CHAPTER 1
INTRODUCTION
2Chapter 1
1.1 BACKGROUND
All onground engineering structures require soil foundation systems to support the
associated structural loads and transfer these loads to the underlying soil. Therefore, it is
essential that checks are made to ensure the underlying soil has sufficient bearing strength
such that collapse does not occur. The capacity of a foundation to carry a given load is
referred to as the ultimate bearing capacity.
In addition to providing adequate load capacity, a routine foundation design process must
include provision for limiting settlement or other movement to tolerable limits.
Two well known cases of bearing capacity failure can be seen in Transcosna Grain Elevator,
Canada Oct. 18, 1913 and Nigata, Japan, 1964 (Figure 1.1).
Factors that affect the bearing capacity of foundations include the soil properties, footing
geometry and the interaction between them. In the past, research into the undrained bearing
capacity of footings has been limited as the interaction of these factors makes the solution
of this problem much more complex. However, the recent advancements in numerical
methods such as finite element method and the rapid increase in computing power mean
that more rigorous solutions to both two dimensional and three dimensional bearing
capacity problems can be found.
The aim of this thesis is to provide comprehensive numerical solutions to three common
bearing capacity problems, namely undrained bearing capacity of surface footings on
layered soils, of embedded footings, and of footings on slopes.
The purpose of this chapter is to provide an introduction to the topic of bearing capacity
and an overview of the thesis. In this context, the common types of bearing capacity failure
modes and standard definitions will be presented.
3Chapter 1
Source: http://www.ngdc.noaa.gov/seg/hazard/slideset/1/1_slides.html
s
Figure 1.1 Two well known cases of bearing capacity failure
(a) Transcosna Grain Elevator, Canada Oct. 18, 1913
(b) Nigata, Japan, 1964
West side of foundation sank >7m
Source: Adapted from Day (2001)
4Chapter 1
1.2 BEARING CAPACITY OF FOUNDATIONS
In order to facilitate the discussion in the later chapters, a number of definitions which will
be used in this thesis are presented below.
Bearing capacity is the ability of a soil to safely carry the pressure placed on it from any
engineered structure without undergoing a shear failure with accompanying large
settlements. Applying a bearing pressure which is safe with respect to failure does not
ensure that settlement of the foundation will be within acceptable limits. Therefore,
settlement analysis should generally be performed since most structures are sensitive to
excessive settlement (Merifield (2005)).
Ultimate bearing capacity is the intensity of bearing pressure at which the supporting
ground is expected to fail in shear, i.e. a collapse will take place (Whitlow (1995)).
The loads carried by structures include a dead load (overall weight of the structure itself)
and a live load (loads that are not permanently imposed on the structure). Live loads may
include those caused by wind, snow, water flow (on bridges), people, animals and movable
furniture (in buildings), vehicles (on the road, highway and bridges), etc. Every
combination of some or all these loads will affect the soil foundation and therefore must
be considered. However, in this thesis the loads and combination of loads are not studied
as these typically are the responsibility of the structural engineer, not the geotechnical
engineer.
The collapse and failure modes for shallow foundations can be divided into three common
types, namely general shear failure, local shear failure and punching shear failure.
The description of the failure modes presented below is summarized from a number of
sources including Merifield (2005), Whitlow (1995), and Liu & Evett (1998).
General Shear: Figure 1.2(a) illustrates a right side rotation shear failure along a well
defined and continuous slip path which will result in bulging of the soil adjacent to the
foundation. A wedge under the footing goes down, and the soil is pushed to the side laterally
and up. Surcharge above and outside the footing helps hold the block of soil down.
Most bearing capacity failures occur in general shear under stress controlled conditions and
lead to tilting and sudden catastrophic type movement. For example, dense sands and
5Chapter 1
saturated clays loaded rapidly are practically incompressible and may fail in general shear.
After failure, a small increase in stress causes large additional settlement of the footing. The
bulging of surface soil may be evident on the side of the foundation undergoing a shear
failure. In relatively rare cases, some radial tension cracks may be present.
Shear failure has been found to occur more frequently under shallow foundations
supporting silos, tanks, and towers than under conventional buildings. Shear failure usually
occurs on only one side, because soils are not homogeneous and the load is often not
symmetric.
Local shear: Figure 1.2 (b) is a punching−type failure, and is more likely to occur in loose
sands, silty sands, and weak clays. Local shear failure is characterised by a slip path that
is not well defined except immediately beneath the foundation. Failure is not catastrophic
and tilting may be insignificant. Applied loads can continue to increase on the foundation
soil following local shear failure.
Punching Shear: Figure 1.2(c) illustrates punching shear failure. Slip lines do not develop
and little or no bulging occurs at the ground surface. Vertical movement associated with
increased loads causes compression of the soil immediately beneath the foundation.
Vertical settlement may occur suddenly as a series of small movements without visible
collapse or significant tilting. Punching failure is often associated with deep foundation
elements, particularly in loose sands.
Factors affecting the ultimate bearing capacity
The principal factors that influence ultimate bearing capacities are type and strength of soil,
foundation geometry, soil weight in the shear zone, and surcharge. Structural rigidity, and
the contact stress distribution do not greatly influence bearing capacity. Bearing capacity
analysis assumes a uniform contact pressure between the foundation and the underlying
soil.
6Chapter 1
Figure 1.2 Typical failure modes and load−displacement curves for:
(a) general shear failure; (b) local shear failure; (c) punching shear failure. Thecircles indicate various interpretations of failure. (Adapted from Coduto 2001,Vesic 1968, Merifield 2005).
Load
Settl
emen
t
Load
Settl
emen
t
Load
Settl
emen
t
(a)
(b)
(c)
D = 0
D > 0
B
q
D is the embedded depth of footing
D
7Chapter 1
Soil Strength. Many sedimentary soil deposits have an inherent anisotropic structure due
to their common natural deposition in horizontal layers. Other soil deposits such as
saprolites may also exhibit anisotropic properties. The undrained strength of cohesive soil
and friction angle of cohesionless soil will be influenced by the direction of the major
principal stress relative to the direction of deposition.
Using strength parameters determined when the major principal stress is applied in the
direction of deposition, the bearing capacity is calculated for each case as follows:
(1) Cohesive Soil. The bearing capacity of a cohesive soil is proportional to the undrained
soil cohesion cu if the effective friction angle � � 0°.
(2) Cohesionless Soil. The bearing capacity of a cohesionless soil �� and mixed c� � �� soils
increases nonlinearly with increases in the effective friction angle.
Foundation Width. The foundation width influences the ultimate bearing capacity in
cohesionless soil. Foundation width also influences settlement, which is important in
determining the design loads. The theory of elasticity shows that for an ideal soil whose
properties do not change with stress level, the settlement is proportional to the foundation
width.
(1) Cohesive Soil. The ultimate “short term” bearing capacity of a cohesive soil of infinite
depth and constant shear strength is independent of the foundation width.
(2) Cohesionless Soil. The ultimate bearing capacity of a footing placed at the surface of
a cohesionless soil where soil shear strength largely depends on internal friction is directly
proportional to the width of the bearing area.
Foundation Depth. The bearing capacity, particularly that of a cohesionless soil, increases
with foundation depth if the soil is uniform. The bearing capacity is reduced if the
foundation is carried down to a weak stratum.
(1) The bearing capacity of larger footings with a slip path that intersects a rigid stratum
will be greater than that of a smaller footing with a slip path that does not intersect a deeper
rigid stratum.
8Chapter 1
(2) Foundations placed at depths where the structural weight equals the weight of displaced
soil usually assure adequate bearing capacity and only recompression settlement.
Exceptions include structures supported by underconsolidated soil and collapsible soil
subject to wetting.
Soil Weight and Surcharge. Subsurface and surcharge soil weights contribute to bearing
capacity. The depth to the water table influences the subsurface and surcharge soil weights.
Water table depth can vary significantly with time.
1.3 THESIS OUTLINE
The purpose of the thesis is to provide a comprehensive set of solutions for undrained
bearing capacity of foundations. For continuous strip footings (L�B � �), a condition of
plane strain can be assumed, and therefore such a problem can be modelled in two
dimensions.
As footings are typically square, circular, or rectangular in shape, the plane strain
assumption may not apply. For these cases, a three−dimensional study of bearing capacity
should be performed to determine the effect of footing shape.
As an overview, the research presented in this thesis will focus on three main problems:
(1) Undrained bearing capacity of surface footings on layered soils;
(2) Undrained bearing capacity of embedded footings;
(3) Undrained bearing capacity of footings near slopes;
The structure of thesis reflects the three main topics listed above. It is organized as follows:
Chapter 2 presents a basic historical review of the bearing capacity of foundations. Each
topic contains previous experimental investigations and theoretical/numerical analyses.
Chapter 3 presents the results obtained for the undrained bearing capacity of surface
footings on layered clays. Both two−dimensional strip footing and three−dimensional
square and circular footing problems will be discussed. The results will be compared to
previous studies. In addition, the effects of smooth or rough footing/soil interaction will be
also discussed.
9Chapter 1
Chapter 4 focuses on undrained bearing capacity of embedded footings. The soil is
homogeneous, but the footing is placed at some depth from the surface of the soil. Similarly
to chapter 3, both two− and three−dimensional configurations will be considered.
Chapter 5 describes the modelling of the undrained bearing capacity of footings near
slopes. The model is non−symmetric but in plane strain conditions and can be modelled in
two−dimensions.
Chapter 6 concludes the thesis, providing a summary of this study and some
recommendations for further work.
10Chapter 2
CHAPTER 2
HISTORICAL REVIEW
11Chapter 2
2.1 INTRODUCTION
The purpose of this chapter is to summarise previous work on the topic of undrained
bearing capacity of foundations. Particular emphasis will be placed on discussing those
works that are most relevant to the problems analysed in this thesis.
The determination of the bearing capacity of foundations has been developed through both
experimental investigations and numerical/theoretical analyses. In this chapter, previous
research results in these two areas are reviewed.
The bearing capacity of foundations has been investigated for many years. Prandtl (1920)
pioneered research into bearing capacity and showed theoretically that a wedge of material
becomes trapped below a rigid plate when it is subjected to concentric vertical loads.
Terzaghi (1943) applied Prandtl’s theory to a strip footing on soil with the assumption that
the soil is semi−infinite, homogeneous, isotropic, weightless and rigid−plastic. Terzaghi
(1943) presented a bearing capacity theory with equations for calculating the ultimate
bearing capacity of surface footings. These results are still in use today.
The majority of past research has been experimentally based and that is why current design
practices are mostly empirical. Some experimentalists have proposed correlations which
satisfy certain demands of geotechnical engineering. However, in some cases, as the
models become more complex (for example when extended to multi−layered soil, arbitrary
shape footings, etc), difficiencies in the empirical approach become apparent. In these
cases, design practice involves the treatment of the complex problem as a simpler one with
the use of large safety factors to account for uncertainties.
The rapid advancement of numerical methods can be applied to calculate the bearing
capacity of foundations, which provides civil engineers with a clear understanding of
foundation problems; we now have the opportunity to solve very complex geotechnical
engineering problems. Among these numerical/theoretical methods are the upper−bound
and lower−bound methods of limit analysis, finite difference method, boundary element
method, and finite element method. Researchers routinely use these methods to solve very
complex problems including the ultimate bearing capacity of foundations. Fortunately,
computer and software technologies have been developing rapidly such that
numerical/theoretical analyses can be used to solve large problems in a short period of time.
12Chapter 2
In addition, visualisation techniques can display the process and results of the
numerical/theoretical analyses. Importantly, most of the results from numerical/theoretical
analyses can be considered as rigorous.
2.2 PREVIOUS EXPERIMENTAL INVESTIGATIONS
The undrained bearing capacity problem has been researched widely since the first
important discoveries made by Prandtl (1920). The essential role and behaviour of
undrained bearing capacity of foundations has been steadily developing since this time.
Over the last two decades, the property of clay and bearing capacity of clay soil has been
investigated. Most concepts and design guidelines for the undrained bearing capacity of
foundations in use at the moment are based on experimental investigations. These
investigations were typically performed with scale models in a laboratory with very few
full−scale field tests undertaken. Scale model testing has been used more often because
full−scale testing of bearing capacity is costly, time consuming and in most cases
impractical. The precision and depth of the subscale measurement capability of
geotechnical laboratory equipment have been improved in recent decades.
As an overview, this section outlines briefly the important contributions of
experimentalists into the three topics:
(1) Undrained bearing capacity of surface footings on layered soils
(2) Undrained bearing capacity of embedded footings
(3) Undrained bearing capacity of footings near slopes
2.2.1 Undrained bearing capacity of surface footings
This subsection describes the experimental investigations reported by Terzaghi (1943),
Meyerhof (1951), deBeer (1970) and Vesic (1973), Brown and Meyerhof (1969), and Das
and Dallo (1984).
An early approximate solution to bearing capacity was defined as a general shear failure
by Terzaghi (1943). The Terzaghi model is applicable to level strip footings placed on or
near a level ground surface where foundation depth D is less than the minimum width B.
The solutions are based on the Limit Equilibrium Method. Assumptions include the use of
13Chapter 2
a surface footing on soil at plastic equilibrium, and a failure surface similar to the one
shown on Figure 1.2(a). Shear resistance of the soil above the base of an embedded
foundation is not included in the solution. Figure 2.1 illustrates the failure mechanism
assumed by Terzaghi (1943).
Referring to Figure 2.1, the total ultimate bearing capacity can be expressed as:
qu � cNc � qNq � 12�BN� (2.1)
where: c = soil cohesion
� = soil unit weight below the base of the footing
q = soil surcharge at the base level of the footing = �D
Nc, Nq, N� are bearing capacity factors.
WedgeZone
Passive Zone
BD q � �DSurcharge Pressure
Figure 2.1 Geometry of failure surface from Terzaghi’s bearing capacity theory (Adapted from Coduto 2001)
��c
Soil Properties
Radial shear zone
�
Qu � quB
� B
45 � ��2 45 � ��2
Since the founding works of Terzaghi (1943), geotechnical engineers around the world
have considered this as the base−line analysis of the ultimate bearing capacity of
foundations. Equation (2.1) was an approximate solution which uses the superposition
technique to combine the effects of cohesion, surcharge and soil weight.
14Chapter 2
Meyerhof (1951) suggested a general bearing capacity theory with consideration for
correction factors for eccentricity, load inclination, and foundation depth. Not only the soil
below the foundation, but also the influence of the shear strength of soil above the base of
the foundation was investigated.
According to Meyerhof (1951), the ultimate bearing capacity qu is written as:
qu � cNcFcsFcdFci � qNqFqsFqdFqi �12�BN�F�sF�dF�i (2.2)
where: c,� q,� �,� B,� Nc,� Nq,� N� are defined in (2.1)
Fcs, Fqs, F�s are shape factors
Fcd, Fqd, F�d are depth factors
Fci, Fqi, F�i are inclination correction factors
Undrained bearing capacity of a surface footing can be deduced from equation (2.2) by
considering the soil as clay with shear strength and without internal frictional angle, and
expressed as:
qu � cNcFcs (2.3)
For a strip footing on homogeneous clay, the undrained bearing capacity factor is
Nc � � 2 � 5.14, an exact solution, first found by Prandtl, 1920. Studies on square,
rectangular and circular footings on homogeneous clay can be found in the works of deBeer
(1970) and Vesic (1973). They performed a range of tests and suggested the shape factors
in order to calculate the bearing capacity of rectangular, circular and square footings on
homogeneous soil. These shape factors are shown in the Table 2.1.
Shape of footing Fcs Fqs F�s
Strip 1.0 1.0 1.0
Rectangular 1 � BLNq
Nc
1 � BL tan � 1 � 0.4B
L
Circular 1 �Nq
Nc
1 � tan� 0.6
Source: Adapted from Whitlow (1995)
Table 2.1 Shape factors of deBeer (1970) and Vesic (1973)
15Chapter 2
While performing experimental investigations on the bearing capacity of shallow
foundations on a strong sand layer underlaid by soft clay, Das and Dallo (1984) also
researched the homogeneous clay model in the laboratory. In these tests, the three types of
footing, namely strip footing (L�B � �), square footing (L�B � 1), and rectangular
footing (L�B � 2 and 3) were investigated. Clay was packed into a wooden box having
inside dimensions of 24 in. x 24 in. x 30 in. high (0.61m x 0.61m x 0.76m) and model
footings were made of 3/8 in. (9.53mm) aluminium plates. After compacting the clay into
the test box, the average vane shear strength of compacted clay was 251lb�ft2
(12.02KN�m2). The results are in the form of an ultimate bearing capacity qu, so the
bearing capacity factor can be back calculated as:
Nc �qu
Fcs(2.4)
For strip footings Nc � 4.8 which is smaller than Prandtl’s exact solution Nc � 5.14. The
reasons for that were explained by two major factors, namely inaccurate measurement of
cu and local shear failure in the soft clay layer under testing. Other results are Nc � 5.71
for L�B � 1, Nc � 5.39 for L�B � 2 and Nc � 5.11 for L�B � 3. With these results,
a chart for bearing capacity of strip footings and rectangular footings for a range of L�B
varying from 0 to 1 was proposed.
However, a foundation usually consists of more than two layers. Brown and Meyerhof
(1969) made a significant contribution to the theory of the bearing capacity of layered
clays. They used a scale model consisting of two layers of different strength clays of
varying thickness. The experiments were carried out in terms of total stresses. For the
strong over weak clay layer case, failure occurs as the footing punches through the top layer
with full development of bearing capacity of the lower layer. For the weak layer over strong
layer case, failure occurs mainly by squeezing of the soft layer between the footing and
stiffer lower layer, with some interaction between the layers as the strength ratio
approaches unity. However, there were some limitations of the study. Firstly, all the studies
were carried out in terms of undrained shear strength of clay, using total stresses or � � 0
analyses. Secondly, the studies were confined to surface loadings using rigid strip and
circular footings with rough bases. Thirdly, only one type of clay was used.
16Chapter 2
Brown and Meyerhof (1969) also performed tests on strip and circular footing on
homogeneous clay. From the strip footing test results, the bearing capacity factor, Nc, was
calculated as 5.14, while from the circular footing tests the bearing capacity factor, Nc was
calculated as 6.05. In addition, the general load−settlement behaviour and failure zones
were observed. The typical load−settlement curves have the form of local shear failure
similar to the curve shown in Figure 1.2b.
For the case of footings on strong clay overlying weak clay, the bearing capacity factor was
found to be a function of only the ratio of the top layer thickness to the footing width or
diameter, H�B or H�2R which varied from 0.5 to 3.0. One limitation of this study is that
the shear strength of the top layer (Ct) was four times the shear strength of the bottom layer
(Cb) for all the tests. The point of failure was clearly visible at an average penetration of
16% of the footing width for strip footings, and 7% of the footing diameter for circular
footing tests. Brown and Meyerhof (1969) produced equations for the modified bearing
capacity factors for strip and circular footings as Nms and Nmc respectively. They were a
function of the variables H�B or H�2R and Cb�Ct.
qf � CtNms for strip footing (2.5)
qf � CtNmc for circular footing (2.6)
Nms � 1.5 HB� 5.14
Cb
Ct� 5.14 (2.7)
Nmc � 3.0 H2R
� 6.05Cb
Ct� 6.05 (2.8)
For the cases of footings on weak clay over strong clay, the test configurations were similar
to the cases of footings on strong clay overlying weak clay. Again, the point of failure was
noticed. At the point of failure, average penetrations for the complete series of strip and
circular footing tests were 6% and 3% of the footing width respectively. In these cases,
Brown and Meyerhof (1969) also proposed an equation for estimating the modified bearing
capacity factors Nms and Nmc, but both of these modified bearing capacity factors were
independent of the ratio of shear strength between the two layers:
Nms � 4.14 � 0.5 BH
(2.9)
17Chapter 2
Nmc � 5.05 � 0.33 2RH
(2.10)
For rectangular footings, Brown and Meyerhof (1969) proposed an equation for the
modified bearing capacity by modifying the equations for strip and circular footings and
including the ratio of width to length of the rectangular footing:
Nmr � NmcBL� Nms1 � B
L (2.11)
where Nmc and Nms are obtained from their figures for modified bearing capacity factors.
Equation (2.11) can be used for both cases of strong over weak and weak over strong clay
layers. The equations (2.5) to (2.11) are being used in practice.
Layered soils are very common in geotechnical engineering, and still constitute a huge
problem for researchers throughout the world. Meyerhof (1974), Meyerhof and Hanna
(1978), Das and Dallo (1984), Michalowski and Shi (1995) tested the model of sand over
clay layers.
Meyerhof (1974) investigated a model of both dense sand over weak clay, and loose sand
on stiff clay. He found that the ultimate bearing capacity of footings on sand layers
overlying clay can be expressed by a punching shear coefficient for the case of dense sand
on weak clay, and by a modified bearing capacity coefficient or an empirical interaction
relationship for the case of loose sand on stiff clay. The theory of bearing capacity, and
Meyerhof’s (1974) test also showed that the influence of the sand layer thickness beneath
the footing depends mainly on the bearing capacity ratio of the clay to the sand, the friction
angle of the sand, and the shape and depth of the foundation. For stiff sand overlying weak
clay model, this result can be seen as a development of punching theory. Following this
idea, and the result of Brown and Meyerhof (1969), a division between full, partial
punching and general shear failure have been found by Merifield et al (1999) for strip
footing, and Merifield and Nguyen (2006) for strip, square and circular footing by finite
element analyses. This division is discussed in the Section 3.4.4 and Figure 3.9.
Das and Dallo (1984) studied the bearing capacity of strip, square and rectangular footings
by experimenting with sand over clay. However, the properties of the layered sand/clay
were the same in all tests. The thickness of the top layer and the shape of the footing were
18Chapter 2
changed. The model test results were compared with the theory of bearing capacity. In
general the experimental and theoretical results are quite close to each other. In the tests
for square footings, only the cases of ratio H�B of 1/3 to 1.95 were investigated. For
rectangular footings where L�B � 2 Das and Dallo (1984) studied only H�B=1/3 to 2.5.
And for L�B � 3 only H�B ratios of 0.5 to 2.9 were investigated. For strip footings ratios
of H�B � 1 to 3.57 were studied.
Indeed, it is very difficult to test all the different layered configurations with varying soil
properties. The alternative is to construct numerical models, and run a virtual simulation.
2.2.2 Undrained bearing capacity of embedded footings
This subsection describes the experimental investigations and formulas of undrained
bearing capacity of embedded footings reported by Terzaghi (1943), Meyerhof (1951), and
Skempton (1951).
Most foundation structures use embedded footings because they have a higher bearing
capacity than surface footings, all other things being equal. This can be seen in equation
(2.1) by Terzaghi (1943) or equation (2.2) by Meyerhof (1951), where the bearing capacity
factors Nc of embedded footings is larger than that of surface footings.
For undrained bearing capacity of foundations embedded in clay, where shape and depth
factors are considered, equation (2.2) becomes:
qu�FcsFcdcuNc � �DNq (2.12)
where: Fcs = the shape factor;
Fcd = the depth factor;
cu= the undrained strength of the soil;
�DNq= the surcharge.
The Terzaghi bearing capacity factors ( Nc, Nq, N�) are functions of the soil angle of
internal friction (�). For a strip footing on undrained soil, � =0, Nc=2+, Nq=1.
However, there is some uncertainty regarding the values of depth and shape factors Fcs and
Fcd for square, circular and rectangular footings. Meyerhof (1951) proposed the following
shape and depth factor for undrained soil (�=0):
19Chapter 2
Fcs � 1 � 0.2BL (2.13)
Fcd � 1 � 0.2Df
B (2.14)
Skempton (1951) also suggested that for undrained clay, the basic form of Terzaghi’s
equation should be used but the bearing capacity factor Nc should be modified to include
the effects of depth and shape as follows:
Nc � 5.141 � 0.2 BL�1 � 0.053 D
B � (2.15)
=5.14FcsFcd
with maximum values when D�B � 4.0 as follows:
when B�L � 0, Nc � 7.5 (i.e. strip footing);
when B�L � 1.0, Nc � 9.0 (i.e. square or circular footing).
These values can be found from Figure 2.2.
4
5
6
7
8
9
10
0 1 3 4 52
Circular or square B/L=1
Strip B/L=0
Nc
D�B
Intermediate
values by
interpolation
Figure 2.2 Skempton’s bearing capacity factor for undrained conditionsNc
20Chapter 2
Equation (2.15) and chart in Figure 2.2 are being used widely in practice.
2.2.3 Undrained bearing capacity of footings near slopes
Footings or foundations are sometimes placed on slopes, adjacent to slopes or near a
proposed excavation. Several civil engineering examples include embankment roads,
highways, railway foundations and abutments of bridges. However, experimental
investigations to ascertain the effect of placing a footing near a slope are very limited. This
subsection describes the experimental investigations of footings on slopes reported by Liu
and Evett (1998).
Liu and Evett’s (1998) study on the bearing capacity of footings on slopes was presented
for both cohesive and cohesionless soil (Figure 2.3). The results include cases for both
footings on slopes, and footings at the top of slopes. If the footings are near or on slopes,
their bearing capacity will be less than if they were on level ground. The ultimate bearing
capacity for strip footings on slopes can be determined from the following equation:
qult � cNcq � 12�BN�q (2.16)
For cohesive soil, (2.16) reduces to:
qult � cNcq (2.17)
where Ncq are bearing capacity factors for footings on slopes and can be determined from
charts in Figure 2.3 of Liu and Evett (1998).
However, these results only apply to the two cases of D�B=0 and 1, and slope stability
factors of Ns=0 to 5 for D�B=0, and Ns=0 for D�B=1 and slope angles of 30°, 60°and
90°. For circular and square footings on slopes, it was assumed that the ratio of their
bearing capacities on slopes to their bearing capacities on level ground are in the same
proportions as the ratio of bearing capacities of continuous footings on slopes to the bearing
capacities of continuous footings on level ground. Hence, their ultimate bearing capacities
can be evaluated by first computing qult by equation (2.16) (i.e., as if the given footing on
a slope were a continuous footing), which is then multiplied by the ratio of qult computed
from the Terzaghi’s equations (2.19) or (2.20) (as if the given circular or square footing
were on level ground) to qult determined from Terzaghi’s equation (2.18) (as if the given
continuous footing was on level ground). This may be expressed in the equation (2.21).
21Chapter 2
Figure 2.3 Bearing capacity for footings on top of slopes
ÉÉÉÉÉÉ
Footing
�°
H
B Foundation
b
D
0 1 2 3 4 5 60
1
2
3
4
8
5
6
7
30°
60°90°
0°
0°
90°60°
30°
90°60°
60°90°
30°
0°
0°30°
Ncq
Ns � 0
Distance of Foundation from Edge of Slope,
Inclination ofSlope, �
Ns � 0
Ns � 4
Ns � 2
Ns � 1
Foundation depth and WidthD/B=0D/B=1
b/B (for ) or b/H (for )Ns � 0
Ns ��Hc = Slope stability factor;
�� � Unit weight of soil
c = Cohesion
22Chapter 2
qult � cNc � �DfNq � 0.5�BN� for continuous footings (width B) (2.18)
qult � 1.2cNc � �DfNq � 0.6�BN� for circular footings (radius R) (2.19)
qult � 1.2cNc � �DfNq � 0.4�BN� for square footings (width B) (2.20)
(qult)ci�or�s�footing�on�slope � (qult)co�footing�on�slope[(qult)ci�or�s�footing�on�level�ground
(qult)co�footing�on�level�ground
] (2.21)
Note: “ci, co or s” footing denotes either circular, continuous or square footing.
These results have been used in practice in many countries throughout the world.
2.3 PREVIOUS THEORETICAL ANALYSES
As well as the experimental investigations already discussed, numerically and theoretical
analyses have also contributed significantly to the understanding of the bearing capacity
of foundations. With the powerful ability of computers many complex models of soil
foundations have been analysed in a short period of time. This is particularly the case for
undrained bearing capacity of footings in cohesive soils.
Nowadays, the trend of research is to validate numerical/theoretical solutions with
experimental investigations. Laboratory experimental investigations are often difficult to
perform, costly, and time consuming making it very hard to obtain a comprehensive set of
results. It can also be difficult to extend from laboratory research to full scale problems with
variable parameters such as geometry, material properties and environmental effects. The
cost and time required for performing laboratory tests on each and every field problem is
very prohibitive. In order to avoid this problem, research can be directed toward
numerical/theoretical models to first discover the fundamentals of a particular problem.
Existing numerical analyses generally assume a condition of plane strain for the case of
undrained bearing capacity of strip footings, and axi−symmetry for the case of undrained
bearing capacity of circular footings. There have been, however, some three−dimensional
solutions for square, rectangular and circular footing problems such as Salgado et al.
(2004), Michalowski et al. (1995) and Merifield and Nguyen (2006).
This section presents a brief historical review of numerical and theoretical analyses for the
three problems in the thesis, namely undrained bearing capacity of surface footings on
23Chapter 2
layered soils, undrained bearing capacity of embedded footings undrained bearing capacity
of footings on slopes.
2.3.1 Undrained bearing capacity of surface footings
Prandtl (1920) developed a plastic equilibrium theory which provided a method for the
determination of the ultimate bearing capacity of strip footing on the surface of a soil
having both cohesion and internal friction. Prandtl’s result for the bearing capacity factor
of strip footings on homogeneous clay is Nc � 2 � � 5.14. This can be considered as
an exact solution.
Using the limit equilibrium method, Terzaghi (1943) proposed the well−known equations
(2.1) for the bearing capacity of foundations. Over the past 60 years, many investigators
have proposed to modify and extend Terzaghi’s method for calculating the bearing capacity
of foundations for more complex models.
This subsection describes the numerical investigations in undrained bearing capacity of
surface footings on the layered soil reported by Reddy and Srinivasan (1967), Davis and
Booker (1973), Griffiths (1982), Merifield et al. (1999) and Salgado et al. (2004).
Reddy and Srinivasan (1967) investigated the bearing capacity of strip footings on layered
clay foundations. In this investigation, they made four basic assumptions, namely that the
potential surface of failure is cylindrical, the coefficient of anisotropy is the same at all
points in the foundation medium, the soil in each layer is either homogeneous with respect
to shear strength or in a given direction the shear strength in each layer varies linearly with
depth and for soil at failure, � � 0 analysis is valid. Reddy and Srinivasan (1967)
considered the shear strength to vary with horizontal or vertical directions or with depth.
The moment equilibrium equation about the centre of potential failure surface must be
satisfied.
Reddy and Srinivasan (1967) considered a range of relative shear strengths between
horizontal and vertical direction K � c2�c1 and between the top of first layer and the top
of the second layer n, n � c�1�c1 � 1 � c�2�c2 � 1,
where c1, c�1 = horizontal shear strengths at the top of the first and second layer respectively;
c2, c�2 = vertical shear strengths at the top of the first and second layer respectively;
24Chapter 2
In practice, the value of K is found to be between about 0.8 and 2.0. The values for n and
the corresponding c�1�c1 are in Table 2.2.
n −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0
c �1�c1
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2
Table 2.2 Values for n and corresponding c�1�c1
Note that n � 0 (or c�1�c1 � c�2�c2 � 1) corresponds to the common case of soft layer
over weak layer, while n � 0 (or c�1�c1 � c�2�c2 � 1) corresponds to the reverse.
There were two cases of layered clays in this investigation. For the first case, the strength
was assumed constant with depth in each layer. For the second case, the strengths vary
linearly with depth in each layer. Both cases used limiting equilibrium of the mass above
the potential surface of rupture, the total disturbing moment about O (centre of the
cylindrical mentioned in basic assumption) will be equal to the total resisting moment about
the same point. Solving the equilibrium moment equations Reddy and Srinivasan (1967)
obtained the bearing capacity of a footing on layered clay q�c2, denoted Nc.
Another parameter affecting the bearing capacity factors of layered clay soil mentioned in
the study of Reddy and Srinivasan (1967) is the thickness of the top layer which has been
expressed by the ratio between thickness of the top layer and the width of footing d�b. In
this investigation for K of 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0 Reddy and Srinivasan (1967)
considered:
(1) d�b=0, 0.2, 0.4, 0.6, 0.8 and 1 for the weak layer over strong layer cases, and
(2) d�b=0, 0.5, 1.0, 1.5, 2.0 and 3.0 for the strong layer over weak layer cases.
All of the bearing capacity of factors Nc for each case were presented graphically.
The case for K=1, c�1�c1 � 1 corresponds to isotropic and homogeneous clay, the bearing
capacity of factor Nc was estimated as 5.60 (higher than the exact solution of Prandtl
(1920), Nc � 5.14). For values of K larger than 1, the value of the ultimate bearing
capacity for the anisotropic medium is considerably smaller than that for the isotropic
medium considering the vertical shear strength to be the same for the two cases. For values
25Chapter 2
of K less than 1, the value of the ultimate bearing capacity for the anisotropic medium is
larger than that for the isotropic medium, the vertical shear strength being considered the
same in the two cases. For the range of anisotropy considered, the ultimate bearing capacity
could be reduced by about 30% or increased by approximately 15% of the ultimate bearing
capacity for the isotropic case.
Davis and Booker (1973) also investigated the effect of increasing the strengths with depth
on the bearing capacity of clays. By means of the theory of plasticity, their results showed
that the rate of increase of cohesion with depth plays the same role as density plays in the
bearing capacity of homogeneous cohesive−frictional soils. Davis and Booker (1973)
considered four cases of non−homogeneous shear strength in the vertical direction
c � c(z), a function of depth z. The soil is assumed purely cohesive and isotropic in the
horizontal direction. Theoretically, they started from an equation of the characteristic lines
s1 and� s2 Figure 2.4, and then the variation in stress state along characteristic lines to
determine the failure.
Figure 2.4 Stress characteristics
x
z
�4
4
s1
s21
characteristic
The stress field and velocity field were obtained to determine the point at which failure
starts. However, in order to solve a system of partial differential equations, a suitable
numerical technique must be used. There were two types of interaction between the
26Chapter 2
footings and the soils investigated in this study, i.e. smooth footings and rough footings.
Both of these cases have the same form of equation for bearing capacity, but with different
dimensionless factor F, i.e. F � FS corresponds to smooth footings while F � FR
corresponds to rough footings (Figure 2.5).
Q
B� F�(2 � )c0 � �B
4� (2.22)
where c0 = cohesive strength on the surface;
� = the rate of increase of cohesion with depth.
The ratio FR�FS showed that roughness increases the bearing capacity by a maximum of
16% and causes no increase at the two limits c0 � 0 and � � 0.
1.0
1.2
1.4
1.6
1.8
2.0
0 4 8 12 16 20 24 28 32 36 40
Figure 2.5 Correction factors F for both rough and smooth footings
Rough FR
Smooth FS
FR�FS
�B�c0
F
The effect of a stiff surface layer, application to embankments and comparison with slip
circular solutions were also investigated.
Griffiths (1982) used the finite element method to study the bearing capacity of footings
on cohesive soil whose strength varies linearly with depth, and the case of two layers of
different strength but within each layer the strength is constant. Griffiths (1982) considered
both rough and smooth footings. In all cases, the bearing pressure mobilised by a given
vertical displacement was obtained by averaging the vertical stress component occurring
in the first row of integrating points below the displaced nodes. Griffiths (1982) used
27Chapter 2
eight−node quadrilateral isoparametric elements with reduced (2−point) Gaussian
quadrature in both the stiffness and relaxation phases of the calculation. Results of
homogeneous soils were compared to the solutions of Terzaghi (1943) and Prandtl (1920)
for the bearing capacity factors Nc, Nq and N�. For clay (� � 0), Griffith’s (1982) bearing
capacity factor Nc was the same as the results of Terzaghi (1943) and Prandtl (1920). For
soil where 0 � � � 35°, the value of Nc from Griffiths (1982) for smooth footings was
lower than that of Terzaghi (1943) and larger than that of Prandtl (1920). For rough
footings the value of Nc from Griffiths (1982) was close to that of Prandtl (1920).
The non−homogeneous two−layer clay system assumed by Griffiths (1982) consisted of
a top and bottom layer with strengths designated as Ct and Cb respectively. Both weak over
strong and strong over weak systems were studied as the ratio Cb�Ct varied from 0.2 to
2. The ultimate bearing capacity is a function of H�B (ratio between thickness of the top
layer and the width of footing) and Cb�Ct. Griffiths (1982) also compared his result (by
finite element method) with the results of Button (1953) who assumed a simple circular
mechanism of failure based upon the upper bound theorem, and the experimental results
of Brown and Meyerhof (1969). With the advantages of the finite element method, Griffiths
(1982) investigated a wide range of problems and determined the bearing capacity and an
adequate stress field at failure, and the ultimate bearing capacity Qult can be obtained:
Qult � CtNct(2.23)
where Nct is bearing capacity factor which can be compared to results of Button (1953) and
Brown and Meyerhof (1969) (Figure 2.6). Griffiths (1982) chose a particular case of
H�B=0.5 for comparison the finite element results with Button (1953) and Brown and
Meyerhof (1969) for a range of Cb�Ct ratio in Figure 2.6. Generally good agreement was
found for other H�B ratios.
28Chapter 2
1
2
3
4
5
6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Figure 2.6 Correction factors for both rough and smooth footings
Circular mechanism
Finite elements
Cb
Ct
Nct
Brown and Meyerhof
HB
� 0.5
Among the rigorous plasticity solutions for the bearing capacity of layered clay is the study
of Merifield et al. (1999). He used numerical limit analysis in conjunction with the upper
and lower bound limit theorems of classical plasticity to obtain a rigorous solution of
undrained bearing capacity of strip footing on two−layered clays. Both methods assume
a perfectly plastic soil model with a Tresca yield criterion and generate large linear
programming problems.
The two−layered clay system adopted by Merifield et al. (1999), consisted of both strong
over weak and weak over strong layers, and was characterised by the ratio of the shear
strength of the top layer to that of the bottom layer cu1�cu2. This ratio was varied from 0.2
to 5, covering most of the practical cases of geotechnical engineering. Another parameter
investigated influencing the bearing capacity of footing on two−layered clay systems is the
ratio of the thickness of top layer to the footing width H�B. Most cases of interest were
covered by varying H�B from 0.125 to 2. The bearing capacity of a shallow strip footing
on two−layered clay without surcharge was expressed as:
qu � cu1N*c (2.24)
where cu1 = the shear strength of the top layer and
N*c = a modified bearing capacity factor which is a function of both H�B and
cu1�cu2; for a homogeneous profile N*c � Nc.
29Chapter 2
qu = ultimate bearing capacity of strip footing obtained from elements under footing
at the state of failure.
The average of the upper and lower bound solutions for N*c were then compared to the
results of the available upper bound solutions of Chen (1975) and results of the
semi−empirical solutions of Meyerhof & Hanna (1978). All of them were expressed in both
tabular and graphical form.
For homogeneous clay, Merifield et al. (1999) obtained an upper and lower bound estimate
of the bearing capacity factor N*c= 4.94 and 5.32 respectively. The average of the upper
and lower bound results produced N*c = 5.13 which is close to the exact well−known
Prandtl’s solution (1920) N*c � 2 � � 5.14. Merifield et al. (1999) compared these
results to N*c=5.53 obtained from the upper bound solution of Chen (1975) and N*
c=5.14
obtained from the semi−empirical solutions of Meyerhof & Hanna (1978).
Salgado et al. (2004) used lower bound and upper bound limit analysis in two− and
three−dimensions for footings in homogeneous clay. In the case of surface footings when
D�B=0, the ratio of net bearing capacity to undrained shear strength, qnetbL�su can be
expressed as a bearing capacity factor Nc, as shown in Table 2.3:
Strip footing Circular footing Square footing
Lower bound Upper bound Lower bound Upper bound Lower bound Upper bound
5.132 5.203 5.856 6.227 5.523 6.221
Table 2.3 Bearing capacity factors Nc of Salgado et al. (2004)
In the case of a footing on a strong over weak clay profile, the results of Merifield et al.
(1999) indicated that, there is a complex relationship between general, local and punching
shear failure and the ratios H�B and cu1�cu2. The limit at which punching shear through
the top layer occurs in the model and local shear failure has been established depending on
H�B and cu1�cu2. Merifield et al. (1999) also showed that the limit H�B>2 at which
failure happens entirely within the top layer, is independent of the ratio cu1�cu2. Chen
(1975) also had a similar limit but Meyerhof and Hanna (1978) had a limit H�B � 2.5.
In the case of a footing on a weak over strong clay profile, Merifield et al. (1999) concluded
that the bearing capacity increases as the relative strength of the bottom layer rises for ratios
30Chapter 2
of H�B � 0.5. There is also a limiting ratio of cu1�cu2 at which time no further increase
in the bearing capacity is achieved as the failure surface becomes fully contained within
the top layer. However this limit increases as the top layer thickness rises. For all values
of H�B �0.5, the failure occurs entirely within the top layer and the bearing capacity is
independent of the strength of the bottom layer, and is given by Prandtl solution (1920)
N*c � 2 � .
The effects of soil−footing interface have been investigated by Merifield et al. (1999). For
a weak over strong clay system where H�B � 0.5, the effect of footing roughness is
important and can lead to a reduction in bearing capacity by as much as 25%. For a weak
over strong clay system where H�B �0.5, and for strong over weak soil profiles, the
bearing capacity is not affected by footing roughness.
The first problem studied in this thesis is the undrained bearing capacity of surface footings
on layered soils, which is a further development of the work of Merifield et al. (1999). The
scope of investigation ( H�B and cu1�cu2) is the same. While Merifield et al. (1999) used
the lower and upper bound limit theorems for strip footings in two−dimensions, this project
uses the finite element method for strip, square, and circular footings in both two− and
three−dimensional space.
2.3.2 Undrained bearing capacity of embedded footings
This subsection describes the numerical investigations of the bearing capacity of embedded
footings reported by Hansen (1969), Hu et al. (1998), Salgado et al. (2004), and Edwards
et al. (2005).
Hansen (1969) investigated the bearing capacity of a vertically and centrally loaded strip
footing placed at a depth D below a horizontal, unloaded surface in homogeneous clay in
the undrained state by the means of the theory of plasticity for ideal rigid−plastic materials
(Figure 2.7a). Hansen (1969) also assumed the soil is weightless (� � 0), no contact at the
vertical soil−footing interface, and shear strength is constant within the soil medium. The
scope of the investigations is limited to relatively shallow footings (0 � D�B � 2). First
of all, Hansen (1969) proposed one of the simplest possible rupture figures, consisting of
a kinematically admissible displacement field, two radial zones with straight radial slip
lines and two line ruptures (Figure 2.7b). The bearing capacity was expressed as a ratio of
31Chapter 2
Q�cB as a function of D�B as curve No.1 on Figure 2.8. Hansen (1969) could calculate
the radius R of radial zone and length of straight line in the failure zone. Applying the
lower−bound theorem, two different solutions for bearing capacity in the form of ratio
Q�cB as a function of D�B were found.
a)
c)
d) e)
Figure 2.7 Bearing capacity problems and solutions
a) Bearing capacity problem
b) Rupture figure with simple radial zones
c) Rupture figure with generalized radial zones
d) Rupture figure with augmented radial zones
e) Rupture figure for mathematically correct solution
b)
Q
D
D D
DD
B B
BB
B
T
T
T
T
A
AA
A
RR
R
PP
B
B
CD
B
B
CD
CL
CLCL
CL CL
Secondly Hansen (1969) had generalized radial zones which yielded better
approximations. The straight line now is replaced by a curve (Figure 2.7c). An
32Chapter 2
upper−bound solution satisfying all equilibrium conditions, and nowhere exceeded failure
conditions is to be found. All the results of the bearing capacity are also expressed as Q�cB
(function of D�B) in the same chart for both models (curve No.2 on Figure 2.8).
The third model is simple augmented radial zones because the first model is kinematically
admissible but on the unsafe side, mainly because of the strong singularity at the point T
on Figure 2.7d. The results of bearing capacity were shown on the curve No.3 on Figure 2.8
The last case is the mathematically correct solution by replacing the simple radial zone with
a generalized zone (Figure 2.7e). The bearing capacity is shown as curve No. 4 on
Figure 2.8 and it corresponds with equation (2.25) for use in practice.
Q
cB� � 2 � 0.533[ 1 � 1.75D�B � 1] (2.25)
0 1.0 2.00
1.0
2.0
3.0
1
2
3
4Q
cB� ( � 2)
DB
Figure 2.8 Bearing capacity variation with depth from Hansen (1969)
Salgado et al. (2004) studied the two− and three−dimensional bearing capacity of strip,
square, circular and rectangular foundations in clay using finite element limit analysis. The
33Chapter 2
results of the analyses are used to propose more rigorous values of the shape and depth
factors for foundations in clay. The shape and depth factors are determined by computing
the bearing capacities of footings of various geometries placed at various embedment
depths D of footings from 0 to 5 times of footing width or diameter. Salgado et al. (2004)
used numerical limit analysis formulations based on the lower−bound and upper−bound
theorems of plasticity (Hill (1951) and Drucker et al. (1951, 1952)) as a tool for all two and
three−dimensional bearing capacity problems. In order to increase the accuracy of
computed depth factors for the three−dimensional problems and reduce the computation
time, the models have exploited symmetry. Sectors of 15°, 45° and 90° were adopted for
circular, square and rectangular footings respectively. For strip footings, only half of the
footing is modelled. In all these cases the boundary conditions can be easily satisfied
In all two and three−dimensional models, Salgado et al. (2004) considered the space
vertically above the footings as being filled with soil in an attempt to model the real
conditions. However in doing so, they had to assume conditions for the interaction between
the top surface of footings and the above soil mass, i.e. perfectly rough with no separation.
In order to do that, normal hydrostatic pressure was applied to the top mass of the soil above
the footings. The interface between the bottom face of the footings and the soil underneath
the footings was assumed rough by prescribing zero tangential velocity for upper bound
calculations and specifying no particular shear stresses for lower bound calculations. For
strip footing problems, the calculations performed for �D�su � 1 and for weightless soil
show that the bearing capacity of embedded strip foundations is represented exactly by
Terzaghi’s bearing capacity equation (2.26) with Fcs � 1:
qbL,net � qbL � q0 � FcsFcdNccu (2.26)
where: Nc = a bearing capacity factor;
cu = a representative undrained shear strength;
q0 � �mD is the surcharge at the footing base level;
�m is the saturated unit weight of soil;
D = the distance from the ground to the base of the foundation element;
Fcs = a shape factor;
34Chapter 2
Fcd = a depth factor;
qbL = limit unit load (referred to as the limit unit base resistance) or bearing capacity
of embedment footing;
qbL,net = net limit unit base resistance.
It was clear from the lower bound stress field and upper bound velocity field for both a
surface and deep foundation that deeper foundations mobilise larger volumes of soil,
dissipate more plastic energy and show mechanisms where the stress rotation becomes less
important than for shallow foundations, with a considerable portion of the mechanisms
consisting of vertical slippage of the soil parallel to the sides of the foundation. The larger
D�B ratios are, the more work needs to be done by the applied load. In this study, the results
for a strip footing on the surface is very close to Prandtl’s exact solution (closer for lower
bound ( Nc � 5.13)). Salgado et al. (2004) also established depth factors by dividing the
average of the lower bound and upper bound bearing capacity qnetbL
values at the various
ratios of D�B by that for a surface foundation. For circular, square, rectangular foundations
analyses, Salgado et al. (2004) considered both upper− and lower bound solutions and
calculated the average values. Shape factors are withdrawn from dividing the qnetbL
of
circular, square or rectangular footings by the qnetbL
of strip footings at the same D�B value.
They concluded that the shape factors are not constant with depth. This is in contrast to the
assumption of independence of shape and depth factors implied by traditional expressions.
From the results of limit analysis, Salgado et al. (2004) proposed an equation for depth
factor of square, circular and rectangular footings:
Fcd � 1 � 0.27 DB
(2.27)
They also used the depth factor equation to calculate the shape factors sc. And they also
built an equation for the shape factor:
Fcs � 1 � C1BL� C2
DB
(2.28)
as a function of B�L and D�B with constants C1 and C2.
35Chapter 2
Deep foundations were also investigated and indicated that for D�B> 1, qnetbL�Fcs>9 while
traditionally it has been taken as 9. For example, for D�B=5, qnetbL�Fcs is at least equal to
11, the value of the lower bound, and possibly as high as 13.7. It is possible that qnetbL�Fcs
would continue to increase with increasing D�B beyond D�B=5. All computations were
performed for a rough soil−footing interface. Salgado et al. (2004) concluded that this is
not often realized, because the safety factor that is used in current design practice accounts
for various sources of uncertainty including those in the analysis, soil properties, load and
boundary conditions or construction uncertainties. The results reduce the uncertainties
with respect to bearing capacity equation, which can lead to lower safety factors.
The second problem this study addresses is the same as modelled by Salgado et al. (2004)
but by a different method. While Salgado et al. used lower−bound and upper−bound
theorems of limit analysis, this study uses the displacement finite element method.
Recently, Edwards et al, (2005) carried out investigations about depth factors for undrained
bearing capacity, by small−strain finite element analysis of embedded strip and circular
foundations using the Imperial College Finite Element Program (ICFEP) (Potts &
Zdravkovic, 1999). In their studies, the D/B ratio was varied from 0 to 4 with every 0.25
as D�B � 1, and every 0.5 as 1 � D�B � 4. For both strip and circular footing
problems, and because of the symmetry in geometry and loading conditions, only half of
the domain was discretized in two−dimensional solutions, using eight−noded quadrilateral
elements. The boundary condition adopted was vertical movement only for the embedded
footings. The footings/soil interactions have both a rough and smooth interface. The soil
was modelled using the Tresca constitutive model, with a constant undrained strength with
depth, su, equal to 50 kPa, a Young’s modulus, E, of 105 kPa and a Poisson’s ratio, , of
0.499. The soil depth of the model underneath the footing is 5B and the width of the model
is 7.5B which is large enough as not to influence the performance of the model. A uniform
vertical displacement was applied to the horizontal surface of the footing. From this
surface, the total reaction force from soil mass is output as a net bearing force, which is
equal to FcsFcdNccuA.
36Chapter 2
Edwards et al, (2005) gathered the results and compared them with previous studies. For
strip ( Nc) and circular surface footings they found out Nc =5.18 (FcsNc) is very close
(+0.8%) to the exact 2 � solution of Prandtl 1920.
The result of FcsNc=6.09 for circular footing is also close (+0.6%) to the 6.05 solution of
Eason & Sheild (1960). The embedded strip footing results were compared to lower bound
and upper bound solutions of Salgado et al. (2004), the bearing capacity equation of
Skempton (1951) in the same chart of dcNc depending on D�B. The embedded circular
footing results were compared to the solutions of Salgado et al. (2004), Houslby & Martin
(2003), Martin (2001) and Skempton (1951) in the same chart of FcsFcdNc depending on
D�B (from 0 to 4).
The depth factors of Edwards et al (2005) were obtained by dividing the bearing capacities
of footings at depth by that for surface footings. They concluded that the shape factor Fcs
is unaffected by depth, while Salgado et al. (2004) concluded that the depth factor derived
for strip footing applies to all footing shapes and to use this to determine how the shape
factor Fcs varied with depth.
Hu and Randolph (1999) investigated the bearing response of skirted foundation on
non−homogeneous soil numerically, analytically and physically with the offshore sediment
simulated as a cohesive soil with strength increasing linearly with depth. In the numerical
analysis, the h−adaptive FEM was adopted to provide an optimal mesh, in which a
strain−superconvergent patch recovery error estimator and mesh refinement with
subdivision concept are used. Hu and Randolph (1999) present two separate studies of
circular skirted foundations on non−homogeneous soil, consisting of a bearing−capacity
study and a large penetration study. The bearing capacity of the foundation is studied with
the degree of non−homogeneity (kD�suo) (where k =strength gradient, D=Diameter or
width of foundation, suo=undrained shear strength at the level of skirt tip) of soil up to 30,
different skirt roughness and skirt depth up to five times the foundation diameter
(i.e., Df�D � 5) ( Df = the skirt penetration depth, D=Diameter or width of foundation)
using an h−adaptive FEM and extended upper−bound method. In small strain analysis, an
optimal mesh is first generated using the h−adaptive refinement strategy, and that mesh is
then used for the bearing−capacity analysis. In large strain analysis, the original (refined)
mesh is updated periodically through the analysis, using the h−adaptive refinement strategy
37Chapter 2
at each stage of remeshing. In all of the FE simulations, the small strain analysis was
implemented in the AFENA FE package, which was developed at the Geotechnical
Research Centre at the University of Sydney (Carter and Balaam 1990). Soil was modelled
as simple−elastic perfect−plastic, taking Poisson’s ratio n = 0.49, stiffness ratio E�su = 500
(E is Young’s modulus), and friction � and dilation � angles equal to zero. The Tresca yield
criterion with associated flow rule was adopted because of the undrained conditions. Hu
and Randolph (1999) investigated the bearing capacities of circular foundations with
embedment up to Df�D =5 for both smooth and rough sides, for a displacement of 0.3D
on homogeneous undrained clay. Solution can be seen as:
qu
su
excluding�side�friction� qu
su
rough�side� 4
Df
D(2.29)
It was apparent that the main difference in the bearing capacity between rough and smooth
sides is due to side friction, and that the base capacities of both agree very closely once the
skirt depth exceeds Df�D=1. Also, the bearing capacity reaches a constant of qu�su=9.9
(for a displacement of 0.3D) when Df�D is >2 (Figure 2.9).
qu
su
Df
D
Figure 2.9 Bearing capacity of circular foundation on homogeneous soil from Hu & Randolph (1999)
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Rough sideRough side excluding side frictionSmooth side
38Chapter 2
2.3.3 Undrained bearing capacity of footings near slopes
This subsection describes the numerical investigations of bearing capacity of footings near
slopes reported by Saran et al. (1989), Narita and Yamaguchi (1990), and Shiau et al.
(2008).
As can be seen in practice, a number of civil engineering structures require the stability of
footings placed near an existing slope. Among these are abutments of bridges, foundations
of roads and highways on the embankment. The bearing capacity of strip footings on slopes
is still an important problem in geotechnical engineering. There are two types of failure
modes for a footing on a slope, namely stability of the slope itself and bearing capacity
failure of the footing. Using numerical analyses, the problem of bearing capacity of
footings on slopes has been studied.
The bearing capacity of footings adjacent to slopes was investigated by Saran et al. (1989).
In their studies, analytical solutions have been developed for obtaining the ultimate bearing
capacity using limit equilibrium and limit analysis approaches. One−sided rupture/failure
on the side of slopes was assumed and partial mobilization was considered on the side of
level ground. Saran et al. (1989) mentioned two types of bearing capacity of slopes, namely
foundation failure and overall stability of the slopes. The foundation was placed on the edge
of the slope and up to three times the footing width away from the slope crest. In addition,
embedment ratios ( D�B) from 0 to 1 were considered. Slope angles of 5°, 10°, 20° and
30° were analysed. In the cases of noncohesive soils, Saran et al. (1989) concluded the
bearing capacity is always governed by bearing capacity failure, while in cohesive material
the stability of the foundation may be dictated by overall slope stability. In the limit
equilibrium analysis, assumptions similar to Terzaghi were made regarding the failure
mechanism shape, with the only exception being the centre of the logarithmic spiral
(Figure 2.1). The bearing capacity expression is then developed by considering the
equilibrium of the elastic wedge underneath of footing. The ultimate bearing capacity was
obtained using the principle of superposition and expressed by introducing bearing
capacity factors. Saran et al. (1989) compared their results with those by Lundgren and
Mortenson (1953) and Saran (1969). In the limit analysis study of Saran et al. (1989), some
basic assumptions have been made as follows: the soil mass is ideally plastic, the failure
mechanism is the same as that adopted in the limit equilibrium analysis, and the associated
39Chapter 2
flow rule is observed. The bearing capacity equations were obtained theoretically by
computations of bearing capacity factors like in limit equilibrium analysis.
The undrained bearing capacity of footings on slopes and seismic bearing capacity of
footings on slopes were studied numerically/theoretically by J. Kumar and Kumar (2003),
Kumar et al. (2003), and Askari et al (2003). However, the mechanism of bearing capacity
is different when lateral accelerations are included.
Narita and Yamaguchi (1990) investigated bearing capacity of strip foundations placed on
the level ground on the top of slopes (Figure 2.10). Comparisons are made with other
analytical and experimental results to examine applicability of the method to practical
problems. It is revealed that the log−spiral analysis somewhat overestimates bearing
capacity values as compared to other solutions, the errors involved being around 20% at
maximum. Also noticed is a relatively good agreement with experimental results,
especially with the model tests on clay (� � 0), for both the ultimate bearing capacity and
shape of sliding surfaces. Narita and Yamaguchi (1990) investigated two types of failures,
namely toe and slope failure and base failure. Results of this investigation showed that for
purely cohesive � � 0 materials where a log−spiral degenerates into a circle, the
log−spiral solutions become almost equal to the values obtained using Bishop’s method and
are at most 3�5% larger than the upper bound solutions.
�L�B
Bearing capacity, kN/m2
Test Upper bound Log−Spiral
30°
0 28.5 21.5 22.5 (1.05)
0.5 44.4 40.7 42.9 (1.05)
1.0 34.4 35.3 37.2 (1.05)
45°
0 32.2 20.7 21.4 (1.03)
0.5 30.8 27.2 28.2 (1.04)
1.0 38.3 26.5 27.4 (1.03)
60°
0 34.6 27.5 28.0 (1.02)
0.5 44.8 33.4 34.0 (1.02)
1.0 30.3 25.3 25.5 (1.01)
( ): ratio to upper bound solution
Table 2.4 Comparisons with test results for clay (Narita and Yamaguchi (1990))
40Chapter 2
Narita and Yamaguchi (1990) compared the results with test results for clays as shown in
the Table 2.4.
Recently, Shiau et al. (2008) researched the bearing capacity of footings on purely cohesive
slopes. In their studies, a rigorous plasticity solution was obtained by using finite element
numerical upper and lower bound methods. They also showed that for a footing−on−slope
system, ultimate bearing capacity of footings may be governed by either the local
foundation failure or global slope failure. Referring to Figure 2.10, the bearing capacity
depends on a wide range of dimensionless ratios, namely the slope angle �°, L�B, cu��B,
q��B and H�B,
where B = footing width,
H = the slope height and
L = the distance from the edge of the slope to the footing edge,
� = soil unit weight,
cu = shear strength of soil,
q��B = slope surcharge.
p
�B� f�°, L
B,
cu
�B,
q
�B, HB (2.30)
Footing
Foundation
L=�B
H
B
�
q � �Df
cu,��,� � � 0
Figure 2.10 Problems notation of Shiau et al. (2008)
The soil is assumed to be undrained following the Tresca yield criterion with a shear
strength cu (� � 0). In the lower bound finite element analysis, the mesh consists of 1585
41Chapter 2
elements with a total number of 4755 nodes and 2335 discontinuities while in the upper
bound finite element analysis, the mesh consists of 2120 elements with a total number of
6360 nodes and 3131 discontinuities. Unlike meshes in the displacement finite element
method, lower bound mesh discretisation permits several nodes to share the same
coordinates. This extra “degree of freedom”, though increasing the total number of the
problem variables, will improve the solution accuracy. The unknown stress field is then
sought while maximizing an integral of the normal stresses over some part of the boundary.
This integral corresponds to an objective function of a typical mathematical programming
problem. Rough and smooth footings for both lower and upper bound analyses were
investigated.
The model assumes the soil obeys an associated flow rule and three main parameters are
changed, namely �°, L�B and cu��B to obtain a wide range of results. The effect of
surcharge q��B, footing roughness and increasing strength with depth were also
investigated.
For weightless slopes, the undrained bearing capacity of strip footings was close to those
of Davis and Booker (1973) for vertical cuts. Shiau et al. (2008) showed that for L�B � 4,
the vertical cut has no effect on bearing capacity of the footing and the bearing capacity
reduces by a factor of 2.5 as the distance ratio L�B is decreased from four to zero.
For slopes with weight, Shiau’s results showed that the normalised bearing capacity p��B
from upper bound and lower bounds was found to be within 5% of each other. Their average
value is very close to true collapse load which has been bracketed to within � 2.5%.
The ratio of L�B was varied from zero to six and cu��B was varied from zero up to ten.
All the analyses were performed for a model with H�B � 3. The results showed that the
dimensionless bearing capacity p��B decreases linearly with the strength ratio (cu��B)
until becoming non−linear and rapidly approaching zero at a particular value of cu��B
where no feasible solution is available from the numerical analysis.
Researching the effect of the slope angle �°, Shiau et al. (2008) indicated that the footing
capacity decreases as the slope angle �° increases. The distance from the footings to the
crest of the slope L also influences the ultimate bearing capacity. As L�B increases from
0, the bearing capacity also increases and tends to reach a constant value at a certain value
42Chapter 2
of L�B for all values of �°. Shiau et al. (2008) indicated that the effects of footing
roughness can be significant for cases with small values of L�B and high slope angles �°,
an effect up to 30% for � � 90°. By enforcing zero horizontal velocity at the footing−soil
interface, the perfectly rough case produces a higher bearing capacity than the perfectly
smooth case. This amount is 30% in the case of L�B = 0, cu��B= 5.0 and � � 90°. In
general, footings with perfectly rough bases always produce a higher bearing capacity than
those with smooth bases and nearly identical results can be obtained when footings are
located outside the slope influence distance. Shiau et al. (2008) found that the existence of
surcharge q on the surface of slope could make the footing capacity either increase or
decrease, depending on both �°and L�B. The effect of increasing strength with depth was
also investigated by Shiau et al. (2008). They used soft normally consolidated clays whose
undrained strength increased linearly with depth. The rate of change in soil cohesion
� � dcu�dz, to the cohesion at the ground surface, �B�cu0 was chosen as 0, 0.5 and 1.0.
Shiau et al. (2008) showed that H�B=3 is large enough for local failure mechanisms which
can develop without being influenced by the bottom or toe of the slope.
2.4 SUMMARY
After researching historical studies about undrained bearing capacity related to the three
topics in this thesis, the following points can be drawn:
� A great deal of laboratory testing has been performed to predict the ultimate bearing
capacity of foundations. However, the investigations are typically limited in scope.
Unfortunately, results obtained from laboratory testing are typically problem specific
and are difficult to extend to field problems with different material or geometric
parameters.
� There have been several numerical methods for bearing capacity problems so far. Each
problem was solved with certain assumptions and results were compared to laboratory
testing. Very few rigorous numerical studies have been undertaken to determine
bearing capacity behaviour.
� A literature review gives us a clear view of previous achievement and some of the
weaknesses on this work. This thesis contains elements that examine the previous
problem results, but by different methods, and use the power of the computer to solve
some the big problems which could not previously be attempted. Using numerical
43Chapter 2
analysis to determine the effects of influent factors to bearing capacity is one part of
this investigations. The three topics in this thesis related to previously unsolved
problems and can play a significant role in the area of bearing capacity of footings for
geotechnical engineering.
44Chapter 3
CHAPTER 3
UNDRAINED BEARING CAPACITY OF
SURFACE FOOTINGS ON LAYERED SOILS
45Chapter 3
3.1 INTRODUCTION
In this chapter finite element analysis is used to predict the undrained bearing capacity of
strip, square and circular footings resting on layered clays. The soil profile consists of two
clay layers having different thicknesses and properties. The results are compared with
previous solutions for strip footings on layered clays. The bearing capacity behaviour is
discussed and the bearing capacity factors are given for various cases involving a range of
layer thicknesses and properties of the two clay soil layers.
The purpose of this study is to produce three−dimensional solutions for square and circular
footings on two−layered clays using the finite element method. To do this the commercial
software package ABAQUS was used. In addition, two−dimensional analyses for the
bearing capacity of strip footings have been performed for verification and comparison
purposes. The results are an extension of the solution for strip footings on two−layered
clays presented by Merifield et al. (1999).
3.2 PROBLEM DEFINITION
The two and three−dimensional bearing capacity problem to be considered is illustrated in
Figure 3.1. A footing of width B or diameter D rests upon an upper layer of clay with
undrained shear strength cu1 and thickness H. This is underlain by a clay layer of undrained
shear strength cu2 and infinite depth. Symmetry has been exploited for the
three−dimensional analyses and only one quarter of the problem domain has been modelled
as shown in Figure 3.1.
The soil was modelled as an isotropic elasto−perfectly plastic continuum with failure
described by the Mohr–Coulomb failure criterion. The elastic behaviour was described by
a Poisson’s ratio, � � 0.49 and a ratio of Young’s modulus to shear strength of
E�cu � 100 � 500 depending on whether the soil was soft or stiff..
The bearing capacity solution to this problem will be a function of the two ratios H�B and
cu1�cu2. Past research by Merifield et al (1999) indicates that a reduction in bearing
capacity for a strong over weak clay system may occur up to a depth ratio of H�B=2 for
strip footings. In this study solutions have been computed for problems where H�B ranges
from 0.125 to 2 and cu1�cu2 varies from 0.2 to 5. This covers most problems of practical
46Chapter 3
B/2
H
Square FootingUpper layer
Lower layer
B/2D/2
H
Circular FootingUpper layer
D/2
Lower layer
B or D
H
(a) Strip footing
(b) Square footing (c) Circular footing
Figure 3.1 Problem Definition
47Chapter 3
interest. Note that cu1�cu2>1 corresponds to the common case of a strong clay layer over
a weak clay layer, whilst cu1�cu2 <1 corresponds to the reverse.
For the simplest case of a footing on homogeneous clay under undrained conditions
including the effect of footing shape without surcharge, equation (2.1) can be re−written
as follows:
qu � cu1NcFcs (3.1)
where Fcs = shape factor. Empirically, the shape factor Fcs is given by:
Fcs � 1 �BLNq
Nc (3.2)
The shape factor Fcs of Salgado et al (2004) is given as:
Fcs � 1 � 0.12 BL
for square, rectangular and circular footings (3.3)
For the case of a layered soil profile, the bearing capacity is given in equation (2.24)
The value of N*c is to be computed using the results from finite element analyses for each
ratio of H�B , cu1�cu2 and for each footing shape. For a homogeneous profile with
cu1�cu2=1, the modified bearing capacity factor N*c equals the Prandtl solution of (2 � ).
3.3 FINITE ELEMENT MODELLING DETAILS
The finite element software ABAQUS was used for solving this problem. For strip, square
and circular footing problems, the ABAQUS model consisted of two parts, namely the
footing and the soil. This is illustrated in Figure 3.1. Typical boundary conditions are
presented in Figure 3.2.
Typical meshes for the problem of strip and three−dimensional footings are shown in
Figure 3.3. For the strip footing case, the mesh is as shown in Figure 3.3 (a). The mesh
consisted of 6−node modified plane strain triangular elements in Figure 3.4 (a) which were
found to provide the best solution convergence. For the three−dimensional case of square
and circular footings the 10−node modified quadratic tetrahedron element was adopted for
similar reasons (Figure 3.4 (b)). These modified elements are specific to ABAQUS and
have been constructed to reduce “node locking” and to have an unambiguous sign of the
48Chapter 3
Elements in planes have
B/2
H
Square FootingUpper layer
Lower layer
B/2D/2
H
Circular FootingUpper layer
D/2
Lower layer
B/2 or D/2
H
(a) Strip footing
(b) Square footing (c) Circular footing
Figure 3.2 Typical boundary conditions
P1 P2
P3
P4
P5
P1 P2
P3
P4
P5
u1 � 0
1
2
u2 � 0
u1 � 0
u1 � u2 � 0
Footing has u1f � 0,� u2f � � �
u1f � 0,� u2f � � �,� u3f � 0Footing has
2
13
2
13
u1 � u2 � u3 � ur1 � ur2 � ur3 � 0
P3, P4 and� P5
u1 � ur3 � ur2 � 0Elements in plane haveP2
u3 � ur1 � ur2 � 0Elements in plane haveP1
Elements in planes have
u1f � 0,� u2f � � �,� u3f � 0Footing has
u1 � u2 � u3 � ur1 � ur2 � ur3 � 0
P3, P4 and� P5
u1 � ur3 � ur2 � 0Elements in plane haveP2
u3 � ur1 � ur2 � 0Elements in plane haveP1
49Chapter 3
1
2
Footing
Soil
(a) Strip footing
1
2
3
1
2
3
(b) Square footing (c) Circular footing
Figure 3.3 Typical mesh layouts used in the finite element models
1
2
Footing
FootingSoil
Soil
50Chapter 3
contact normal stress that is usually associated with second order elements in contact
analyses. The overall mesh dimensions were selected to ensure that the zones of plastic
shearing and the observed displacement fields were contained within the model boundaries
at all times. The underside of all footings were modelled as perfectly rough by specifying
a “tied” contact constraint at the footing/soil interface.
To determine the collapse load of the footing, displacement defined analyses were
performed where the footing was considered as being perfectly rigid. That is, a uniform
vertical prescribed displacement (� �) was applied to all those nodes on the footing. The
total displacement was applied over a number of substeps and the nodal contact forces
along the footing were summed to compute the equivalent bearing capacity.
A distinct advantage of using the finite element method is that it provides the complete load
deformation response. This can provide insight into general footing behaviour, particularly
in regard to the development of the collapse mechanism and deformation serviceability
issues. By observing the load displacement response, a check can be made to ensure that
the ultimate bearing capacity has been reached and that overall collapse has in fact occurred
(i.e. the load−displacement plot reaches a plateau as shown in Figure 3.5).
a) 6−node modified plane strain
triangular element
b) 10−node modified quadratictetrahedral element
Figure 3.4 Element types used for FE modelling
51Chapter 3
qu
cu1
N*c
�B
����� �D
Figure 3.5 Typical load−displacement curve of footing from Abaqus FE analysis
or
3.4 ABAQUS FINITE ELEMENT ANALYSIS (FEA)
RESULTS
3.4.1 Footing on homogeneous clay cu1 � cu2
As an initial check on the three dimensional finite element model, the bearing capacity
factor for the homogeneous case Nc for square and circular surface footings was calculated
and compared to existing published numerical results. This comparison is summarised in
Table 3.1. As can be seen, the bearing capacity factors from the current finite element study
compare well to that calculated using the widely adopted shape factor in equation (3.2), and
the solutions of Salgado et al (2004). The bearing capacity factor for square footings
( Nc � 5.95) was found to be around 2% below that for circular footings ( Nc � 6.05).
This observation is consistent with the findings of Salgado et al (2004) as shown in
Table 3.1. The bearing capacity factor Nc for strip footings was found to be 5.24, around
2% above the classical Prandtl solution of (2 � ). Table 3.2 compares Prandtl’s (1921)
solution for the bearing capacity factor Nc with those of Griffiths et al. (2002), Merifield
52Chapter 3
et al. (1999) and Salgado et al. (2004). In general the ABAQUS results compare well with
previous estimates of Nc.
FEM−ABAQUS Equation
(3.1)
Equation
(3.2)
Equation
(3.1)
Equation
(3.3)
Salgado et al. 2004
Squarefooting
(SQ)
Circularfooting
(CI)
Square
footing
Circular
footing
Lowerbound
Upperbound
Lowerbound
Upperbound
Nc5.95 6.05 6.14 6.168 5.523 6.221 5.856 6.227
Compared
to SQ 0 −−−− +3.19% +3.66% −7.18% +4.55% −−−− −−−−
Compared
to CI−−−− 0 +1.49% +1.95% −−−− −−−− −3.21% +2.93%
Note: SQ: Square footings, CI: Circular footing, −−−: no need to compare. Compared to
SQ(%)=( Nc−5.95)/5.95, Compared to CI(%)=( Nc−6.05)/6.05
Table 3.1 Comparison of bearing capacity factors Nc for square and circular footings
Prandtl
1921
FEM−
ABAQUS
Griffiths
et al’2002
Merifield et al. 1999 Salgado et al.2004
Lowerbound
Upperbound
Average
Lowerbound
Upperbound
N*c
5.14 5.24 5.423 4.94 5.32 5.13 5.132 5.203
Comparedto Prandtl
0 +1.95% +5.51% −3.89% +3.50% −0.2% −0.16% +1.2%
Note: Positive signifies case when Nc >5.14, negative signifies case when Nc <5.14,
Compared=( Nc −5.14)/5.14.
Table 3.2 Comparison of bearing capacity factors Nc for strip footings
3.4.2 Footing on layered clay cu1 � cu2
The bearing capacity factors N*c (equation (2.24)) for the non−homogeneous case
(cu1 � cu2) for strip, circular and square footings on two−layered clays are presented in
Table 3.3, and shown graphically in Figure 3.6. Also shown in these figures are the bearing
capacity factors for strip footings by Merifield et al (1999). Some general observations will
be mentioned before discussing the results for strong over weak and soft over strong
profiles separately in more detail.
53Chapter 3
The bearing capacity factor N*c was found to be almost identical for square and circular
footings for both the strong over weak and weak over strong layer cases. In all cases the
bearing capacity factor for circular footings was around 1−2% above that of a square
footing. This is shown clearly in Figure 3.6 where the results for square and circular
footings are difficult to distinguish from each other.
As discovered by Merifield et al (1999), all of the analyses herein indicate that a complex
relationship exists between the observed modes of shear failure and the ratios cu1�cu2 and
H�B. The modes of failures can best be described as being either “general shear”, “partial
punching shear”, or “full punching shear” similar to that described by Merifield et al
(1999). Full punching shear (Figure 3.7) is characterised by a vertical separation of the top
layer which then effectively acts as a rigid column of soil that punches through to the
bottom layer. In this case only a small amount of heaving is observed immediately adjacent
to the footing, but significant failure is observed below the upper layer. .
Conversely, only a small vertical separation of the top layer is evident for partial punching
shear. General shear failure is as defined by many authors including Terzaghi (1943). In
this mode of failure we observe well defined shear planes developing and extending to the
surface, and bulging of the soil on both sides of the footing.
When comparing the results for square and circular footings to those for strip footings, in
general it was found that the bearing capacity factors N*c for square and circular footings
were larger than those for strip footings. This is applicable for both the strong over weak
and weak over strong profiles, and was to some extent expected. The only exception to this
observation was for the weak over strong profile case where the upper layer is very thin
compared to the footing width, namely when H�B � 0.25. This is shown clearly in
Figure 3.6(a) where the above trend is reversed and the bearing capacity factors N*c for
square and circular footings were smaller than those for strip footings.
Further three dimensional and axi−symmetric analyses with much finer finite element
meshes confirmed this observation. However, further investigation is required to properly
explain the mechanics of this failure mechanism and the lower than expected collapse load.
54Chapter 3
H/Bcu1�cu2
Value of bearing capacity factor N*c
Upper Bound(Merifield et al.
1999)Finite Element Method − ABAQUS
Strip Footing Strip Footing Square Footing Circular Footing
H/B=0.125
0.20 8.55 8.61 7.96 7.95
0.25 8.55 8.61 7.96 7.95
0.40 8.55 8.60 7.96 7.95
0.50 8.55 8.40 7.96 7.89
0.80 6.36 6.21 6.73 6.85
1.00 5.32 5.24 5.95 6.05
1.25 4.52 4.45 5.11 5.27
1.50 3.93 3.83 4.53 4.66
2.00 3.09 3.03 3.73 3.85
2.50 2.61 2.53 3.21 3.32
4.00 1.82 1.75 2.33 2.41
5.00 1.55 1.48 2.00 2.07
H/B=0.25
0.20 6.52 6.34 6.35 6.36
0.25 6.52 6.34 6.35 6.36
0.40 6.52 6.34 6.35 6.36
0.50 6.52 6.33 6.35 6.36
0.80 6.25 5.95 6.27 6.34
1.00 5.32 5.24 5.95 6.05
1.25 4.6 4.62 5.45 5.59
1.50 4.08 4.05 5.03 5.17
2.00 3.34 3.31 4.39 4.51
2.50 2.88 2.84 3.92 4.02
4.00 2.12 2.06 3.04 3.13
5.00 1.85 1.71 2.70 2.78
H/B=0.375
0.20 5.84 5.78 6.00 6.10
0.25 5.84 5.78 6.00 6.10
0.40 5.84 5.78 6.00 6.10
0.50 5.84 5.76 6.00 6.10
0.80 5.84 5.76 6.00 6.10
1.00 5.32 5.24 5.95 6.05
1.25 4.78 4.77 5.74 5.88
1.50 4.28 4.27 5.46 5.60
2.00 3.65 3.58 4.96 5.10
2.50 3.2 3.13 4.53 4.68
4.00 2.42 2.37 3.67 3.80
5.00 2.13 2.07 3.32 3.44
Table 3.3 Values of bearing capacity factor N*c
55Chapter 3
H/B
Value of bearing capacity factor N*c
cu1�cu2
Upper Bound(Merifield et al.
1999)Finite Element Method −ABAQUS
Strip Footing Strip Footing Square Footing Circular Footing
H/B=0.5
0.20 5.49 5.46 5.96 6.04
0.25 5.49 5.46 5.96 6.04
0.40 5.49 5.46 5.96 6.04
0.50 5.49 5.46 5.96 6.04
0.80 5.49 5.46 5.96 6.04
1.00 5.32 5.24 5.95 6.05
1.25 4.94 4.92 5.94 6.02
1.50 4.48 4.49 5.82 5.90
2.00 3.89 3.86 5.46 5.58
2.50 3.47 3.43 5.08 5.23
4.00 2.74 2.67 4.22 4.39
5.00 2.44 2.36 3.89 4.03
H/B=0.75
0.20 5.36 5.24 5.95 6.03
0.25 5.36 5.24 5.95 6.03
0.40 5.36 5.24 5.95 6.03
0.50 5.36 5.24 5.95 6.03
0.80 5.36 5.24 5.95 6.03
1.00 5.32 5.24 5.95 6.05
1.25 5.2 5.17 5.98 6.05
1.50 4.94 4.87 5.98 6.05
2.00 4.37 4.35 5.96 6.05
2.50 4.01 3.96 5.91 6.03
4.00 3.28 3.22 5.24 5.47
5.00 2.98 2.90 4.94 5.15
H/B=1.0
0.20 5.3 5.27 5.93 6.03
0.25 5.3 5.27 5.93 6.03
0.40 5.3 5.27 5.93 6.03
0.50 5.3 5.27 5.93 6.03
0.80 5.3 5.27 5.93 6.03
1.00 5.32 5.24 5.95 6.05
1.25 5.3 5.24 5.94 6.05
1.50 5.18 5.16 5.94 6.05
2.00 4.82 4.77 5.93 6.06
2.50 4.5 4.44 5.93 6.06
4.00 3.83 3.75 5.86 6.04
5.00 3.54 3.44 5.77 5.94
Table 3.3 (continued) Values of bearing capacity factor N*c
56Chapter 3
H/B
Value of bearing capacity factor N*c
cu1�cu2
Upper Bound(Merifield et al.
1999)Finite Element Method −ABAQUS
Strip Footing Strip Footing Square Footing Circular Footing
H/B=1.5
0.20 5.3 5.25 5.94 6.04
0.25 5.3 5.25 5.94 6.04
0.40 5.3 5.25 5.94 6.04
0.50 5.3 5.25 5.94 6.04
0.80 5.3 5.25 5.94 6.03
1.00 5.32 5.24 5.95 6.05
1.25 5.27 5.24 5.94 6.04
1.50 5.31 5.24 5.94 6.04
2.00 5.31 5.23 5.94 6.04
2.50 5.32 5.19 5.94 6.04
4.00 4.84 4.67 5.94 6.04
5.00 4.56 4.40 5.94 6.03
H/B=2.0
0.20 5.3 5.24 5.96 6.05
0.25 5.3 5.24 5.96 6.05
0.40 5.3 5.24 5.96 6.05
0.50 5.3 5.24 5.96 6.05
0.80 5.3 5.24 5.96 6.05
1.00 5.32 5.24 5.95 6.05
1.25 5.26 5.21 5.96 6.05
1.50 5.26 5.21 5.96 6.05
2.00 5.27 5.21 5.96 6.05
2.50 5.27 5.21 5.96 6.05
4.00 5.32 5.21 5.96 6.05
5.00 5.32 5.13 5.96 6.05
Table 3.3 (continued) Values of bearing capacity factor N*c
57Chapter 3
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 0.25
cu1
cu2
Weak
over
Strong
Strong
over
Weak
(b)
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 0.125
cu1
cu2
N*c
Weak
over
Strong
Strong
over
Weak
(a)
N*c
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
Figure 3.6 Values of bearing capacity factor N*c
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
58Chapter 3
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 0.375
cu1
cu2
N*c
Weak
over
Strong
Strong
over
Weak
(c)
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 0.50
cu1
cu2
N*c
Weak
over
Strong
Strong
over
Weak
(d)
Figure 3.6 (continued) Values of bearing capacity factorN*c
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
59Chapter 3
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 0.75
cu1
cu2
N*c
Weak
over
Strong
Strong
over
Weak
(e)
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 1.00
cu1
cu2
N*c
Weak
over
Strong
Strong
over
Weak
(f)
Figure 3.6 (continued) Values of bearing capacity factorN*c
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
60Chapter 3
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 1.50
cu1
cu2
N*c
Weak
over
Strong
Strong
over
Weak
(g)
0
1
2
3
4
5
6
7
8
9
10
11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
HB
� 2.00
cu1
cu2
N*c
Weak
over
Strong
Strong
over
Weak
(h)
Figure 3.6 (continued) Values of bearing capacity factorN*c
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
ABAQUS − Square footing
ABAQUS −Circular footing
ABAQUS −Strip footing
Upper bound, Merifield et al.
1999−Strip footing
61Chapter 3
No heaving adjacent to footing
No heaving adjacent to footing
Strip footing, H/B=0.50, cu1�cu2 � 5
Strip footing, H/B=1.0, cu1�cu2 � 5
Figure 3.7 Examples of punching shear failure for strip footings
62Chapter 3
3.4.3 2−D Strip footing on layered clay cu1 � cu2
For strip footings on layered clays, the finite element bearing capacity factors compare well
to the upper bound solutions presented by Merifield et al (1999). In general, the
displacement finite element solution presented here is within �2% of Merifield’s solution
(see Figure 3.6).
For the strong over weak case where a moderately strong crust exists
(1 � cu1�cu2 � 2.5), failure is generally caused by partial punching shear. As the depth
of the top crust approaches the footing width B, upward deformations within the bottom
layer become restricted causing an increase in the extent of plastic failure. As the top crust
becomes very strong compared to the bottom layer (cu1�cu2 � 2.5) full punching shear
through the top layer occurs. The very strong top layer then serves to greatly restrict both
lateral and vertical movement of the soil contained in the weak layer below (see
Figure 3.7). This results in the formation of a deep zone of plastic shearing within the
bottom layer.
The results indicate that a reduction in bearing capacity for a strong over weak clay system
occurs up to a depth ratio of H�B � 1.5 � 2. This lower limit is applicable for soil
profiles where cu1�cu2 � 2.5, but for profiles that have a very strong top crust with
cu1�cu2 � 2.5, punching failure through the top layer is likely to occur up to depth ratio
of H�B � 2. For ratios of H�B � 2, failure is contained entirely within the top layer and
is independent of the ratio cu1�cu2. These observations are consistent with those predicted
by Merifield et al (1999).
For the weak over strong case, the finite element results indicate that for ratios of
H�B � 0.5, the bearing capacity increases as the relative strength of the bottom layer
rises. At a limiting ratio of cu1�cu2, no further increase in bearing capacity is achieved as
the general shear failure mechanism becomes fully contained within the top layer. For all
values of H�B � 0.5, the solutions indicate that failure occurs entirely within the top layer
and the bearing capacity is independent of the strength of the bottom layer.
3.4.4 3−D square and circular footings on layered clay
Square and circular footings on weak clay overlying strong clay cu1�cu2 � 1
63Chapter 3
For thin upper layers where H�B � 0.375, N*c increases as the ratio of cu1�cu2 decreases.
That is, as the thin top layer becomes weaker and weaker compared to the underlying strong
layer, the contribution made by the underlying layer to the ultimate bearing capacity
increases. For very thin upper layers ( H�B � 0.125), the increase in the bearing capacity
factor N*c above that for the homogeneous case is significant, as shown in Figure 3.6 (a).
This can be explained by observing the failure mechanisms, such as that shown in
Figure 3.8(a). The displacement vector diagram indicates that, although much of the failure
is contained in the weak upper layer, failure still occurs well into the strong underlying
layer.
For upper layer thicknesses where H�B � 0.375, the stronger underlying layer does not
contribute greatly to the ultimate bearing capacity regardless of the ratio of cu1�cu2. As a
consequence no further increase in bearing capacity is achieved as the failure surface
becomes fully contained within the top layer. The failure mechanisms shown in Figure 3.8
(d − l) confirm this. The bearing capacity factor is equal to that found for the homogeneous
case and is shown as a horizontal line in Figure 3.6. This observation is similar to that found
by Merifield et al (1999) for strip footings where for H�B � 0.5, the limit analysis
solutions indicated that failure occurs entirely within the top layer and the bearing capacity
is independent of the strength of the bottom layer.
Square and circular footings on strong clay overlying weak clay cu1�cu2 � 1
In general it was observed that for a given ratio of H�B � 1, the bearing capacity factor
N*c was found to decrease in a nonlinear manner as the ratio of cu1�cu2 increases
(Figure 3.6). This indicates the failure mechanism must penetrate into the underlying
weaker layer which in turn has the effect of reducing the ultimate bearing capacity. This
is confirmed by the displacement velocity plots shown in Figure 3.8(b),(c) and
Figure 3.8(e),(f). In these figures large displacements and zones of failure are seen to occur
in the underlying weak layer.
As previously mentioned, the ABAQUS finite element results for strip footings and the
limit analysis results of Merifield et al (1999) indicate that a reduction in bearing capacity
for a strong over weak clay system occurs up to a depth ratio of H�B � 2. This limiting
ratio of H�B is greater than that observed for circular and square footings (i.e. H�B � 1).
64Chapter 3
WEAK
STRONG
WEAK
WEAK
STRONG
STRONG
Circular & Square footing,cu1�cu2 � 0.2, H�B � 0.25
N*c � 6.35� for� Square�footing
N*c � 6.36� for� Circular�footing
a)
Circular & Square footing,cu1�cu2 � 2, H�B � 0.25
b)
Circular & Square footing,cu1�cu2 � 5, H�B � 0.25
N*c � 2.70� for� Square�footing
N*c � 2.78� for� Circular�footing
c)
Figure 3.8 Displacement vectors for square and circular footing on layered clay
N*c � 4.39� for� Square�footing
N*c � 4.51� for� Circular�footing
65Chapter 3
Figure 3.8 cont’d Displacement vectors for square & circular footing on layered clay
WEAK
STRONG
WEAK
STRONG
WEAK
STRONG
Circular & Square footing,cu1�cu2 � 0.2, H�B � 0.5
Circular & Square footing,cu1�cu2 � 2, H�B � 0.5
Circular & Square footing,cu1�cu2 � 5, H�B � 0.5
d)
e)
f)
N*c � 5.46� for� Square�footing
N*c � 5.58� for� Circular�footing
N*c � 3.89� for� Square�footing
N*c � 4.03� for� Circular�footing
N*c � 5.96� for� Square�footing
N*c � 6.04� for� Circular�footing
66Chapter 3
Figure 3.8 cont’d Displacement vectors for square & circular footing on layered clay
WEAK
STRONG
STRONG
WEAK
STRONG
WEAK
Circular & Square footing,cu1�cu2 � 0.2, H�B � 1
Circular & Square footing,cu1�cu2 � 2, H�B � 1
Circular & Square footing,cu1�cu2 � 5, H�B � 1
g)
h)
i)
N*c � 5.93� for� Square�footing
N*c � 6.03� for� Circular�footing
N*c � 5.93� for� Square�footing
N*c � 6.06� for� Circular�footing
N*c � 5.77� for� Square�footing
N*c � 5.94� for� Circular�footing
67Chapter 3
Figure 3.8 cont’d Displacement vectors for square & circular footing on layered clay
WEAK
STRONG
WEAK
STRONGCircular & Square footing,
cu1�cu2 � 0.2, H�B � 1.5
Circular & Square footing,cu1�cu2 � 5, H�B � 1.5
Circular & Square footing,cu1�cu2 � 0.2 � 5, H�B � 2
k)
j)
l)
N*c � 5.94� for� Square�footing
N*c � 6.04� for� Circular�footing
N*c � 5.96� for� Square�footing
N*c � 6.05� for� Circular�footing
N*c � 5.94� for� Square�footing
N*c � 6.03� for� Circular�footing
68Chapter 3
This behavioural difference is most obvious in Figure 3.6(e) where the bearing capacity
factor N*c for a strip footing is well below that of a circular or square footing on the same
layered profile.
Punching shear failure is a common mechanism observed in the case of strong over weak
profiles. Full punching shear failure was generally observed when cu1�cu2 � 2.5. For
these cases, the top layer acts as rigid column of soil that pushes through the top layer into
the underlying weak soil and a significant zone of failure is therefore apparent in the lower
layer. Full punching shear failure is highlighted in Figure 3.8(c) and Figure 3.8(f). Based
on the observed displacement vectors and stress contours, some guidance is provided in
Figure 3.9 as a means of distinguishing the likely mode of failure for circular and square
footings on a strong over weak profile. A division between full, partial punching and
general shear failure depending on both ratio H�B and cu1�cu2 is provided in Figure 3.9.
1 1.5 2 2.5 3 3.5 4 4.5 50
0.25
0.5
0.75
1ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ
1.25
ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ
Partial Punching Shear Failure
General Shear Failure
Full Punching Shear Failure
Figure 3.9 Division between full, partial punching and general shear failure
HB
cu1cu2
69Chapter 3
3.4.5 Effect of footing roughness
The results of analyses with a perfectly smooth footing/soil interface indicate that for a
strong over weak clay profile the soil/footing interface strength has little or no effect on
the calculated bearing capacity. For strip footings the reduction in the bearing capacity was
� 4% whilst for square and circular footings the reduction was less than 1%. Similarly
for a weak over strong clay system where H�B � 0.5, the bearing capacity does not vary
greatly with footing roughness (i.e. � 4% for all footing shapes). This agrees with the
conclusions made by Merifield et al (1999).
For square, circular and strip footings on a weak over strong clay system where
H�B � 0.5, a perfectly smooth soil/footing interface serves to reduce the bearing capacity
by up to 26%. The greatest reduction occurs when H�B � 0.125 and falls quickly to
around 6% for H�B � 0.5.
3.5 CONCLUSION
The bearing capacity of strip, square and circular footings on two−layered clays using the
finite element method has been investigated. The results obtained have been presented in
terms of a modified bearing capacity factor N*c in both graphical and tabular form to
facilitate their use in solving practical design problems.
The following conclusions can be made based on the finite element results:
� For homogeneous soil profiles, the finite element bearing capacity and shape factors
for square and circular footings compare well to previously reported finite element and
numerical limit analysis solutions.
� For strip footings on layered clays, the finite element bearing capacity factors compare
well to the numerical upper bound limit analysis solutions presented by Merifield et
al (1999). In general, the finite element solution presented here are within � 2% of
solution of Merifield et al (1999).
� For a weak over strong clay system where the ratio of H�B � 0.375, the stronger
underlying layer does not contribute greatly to the ultimate bearing capacity regardless
of the ratio of cu1�cu2. As a consequence no further increase in bearing capacity is
achieved as the failure surface becomes fully contained within the top layer. When
70Chapter 3
H�B � 0.375 the bearing capacity factor N*c increases as the ratio of cu1�cu2
decreases. For very thin upper layers ( H�B � 0.125), the increase in the bearing
capacity factor N*c above that for the homogeneous case is significant.
� For both square and circular footings, a reduction in bearing capacity for a strong over
weak clay system occurs up to a depth ratio of H�B � 1. For depth ratios of
H�B � 1, failure is likely to be fully contained within the top layer and the bearing
capacity factors for square ( N*c =5.95) and circular footings ( N*
c =6.05) on a
homogeneous profile can be used.
71Chapter 4
CHAPTER 4
UNDRAINED BEARING CAPACITY OF
EMBEDDED FOOTINGS
72Chapter 4
4.1 INTRODUCTION
Undrained bearing capacity of embedded footings is an important issue when estimating
bearing capacity. In this chapter the ultimate undrained bearing capacity of embedded
footings is investigated. Consideration has been given to the effect of embedded footing
depth for strip, square, circular and selected rectangular footings. The bulk of the results
are for cases where: (i) rough interfaces exist between the footings and soil; (ii) the footing
has the same shape from the toe to the ground surface; and (iii) no friction operates between
the side of the footing and the soil.
To begin with, the problem of undrained bearing capacity of embedded footings is defined
in terms of soil strength and problem geometry. A number of numerical solutions for the
embedment capacity are then presented which are based on finite element methods. These
calculations are compared with the numerical limit analysis results of Salgado et al (2004)
in a subsequent section of this chapter.
4.2 PROBLEM DEFINITION
A general layout of the problems to be analysed is shown in Figure 4.1 and Figure 4.2.
The system includes a footing and a soil foundation. The footing is placed at an embedment
depth D from the surface. The clay soil foundation has an undrained shear strength cu and
infinite depth. The geometry of each type of footing is characterised as follows:
− Strip footing has width B;
− Square footing has side length L;
− Rectangular footing has width B and length L; and
− Circular footing has diameter B.
In this thesis, equation (2.2) is used to obtain modified bearing capacity factors for a soil
without unit weight. For purely cohesive clay (�u � 0), equation (2.2) therefore reduces
to:
qu �Qu
A� cuNcFcsFcd (4.1)
The purpose of this study is to propose bearing capacity, shape, and depth factors for strip,
square, circular and rectangular footings at some depth in clay using the finite element
method. Once again, the commercial software package ABAQUS was used.
73Chapter 4
Symmetry has been exploited for the problems. Only one quarter of the problem domain
has been modelled for three−dimensional analyses and one half for two−dimensional
analyses (Figure 4.1 and Figure 4.2).
The soil was modelled as an isotropic elasto−perfectly plastic continuum with failure
described by the Mohr–Coulomb yield criterion. The elastic behaviour was described by
a Poisson’s ratio � � 0.48, and a ratio of Young’s modulus to shear strength of
E�cu � 500. The soil is treated as weightless. The interactions between the soil and the
bottom of the footings are characterized as rough.
For this problem the undrained bearing capacity of embedded footings will be a function
of the ratio D�B and/or D�L.
In this thesis, solutions have been computed for problems where D�B ranges from 0 to 5.
In the rectangular case, the additional parameter B�L varies from 0.2 to 0.5. These ranges
cover most problems of practical interest.
The output of the ABAQUS analyses is the total reaction Qu force on the footing from
which the ultimate bearing capacity qu can be calculated. In addition, the depth factor Fcd
and shape factors Fcs is obtained via equation (4.1).
4.3 FINITE ELEMENT MODELLING DETAILS
The ABAQUS model consisted of two parts, namely the footing and the soil. This is
illustrated in Figure 4.1 and Figure 4.2. At the interface between the footing and the soil,
the elements representing the footing and those representing the soil have the same
topology. The applied boundary conditions are presented in Figure 4.3 and Figure 4.4.
Typical meshes for the problem of the two−dimensional strip, and the three−dimensional
square and circular footings are shown in Figure 4.5, those of the three−dimensional
rectangular footing are presented in Figure 4.6. For the strip footing case, the mesh
(Figure 4.5(a)) consisted of 8−node plane strain quadrilateral, hybrid, linear pressure,
reduced integration elements (CPE8RH) as shown in Figure 4.7(a) because this element
was found to provide the best solution convergence. For the three−dimensional case of
square, rectangular and circular footings, the 10−node modified quadratic tetrahedron
element (C3D10M) was adopted for similar reasons Figure 4.7(b).
74Chapter 4
Figure 4.1 Problem definition for strip, square and circular footings
(a) Strip footing
(c) Circular footing(b) Square footing
D
D
B/2 B/2
B/2B/2
B
D
Footing
Footing
Footing
Soil
Soil
Soil
1
2
31
2
3
1
2
75Chapter 4
Footing Footing
FootingFooting
DD
DD
B/2B/2
L/2L/2
B/2 L/2
L/2B/2
Figure 4.2 Problem definition for rectangular footings
(a) Rectangular footing, B/L=0.50
(d) Rectangular footing, B/L=0.20(c) Rectangular footing, B/L=0.25
(b) Rectangular footing, B/L=0.33
2
31
1
2
3
2
31
1
2
3
SoilSoil
SoilSoil
76Chapter 4
Figure 4.3 Typical boundary conditions for strip, square, and circular footings
(a) Strip footing
(c) Circular footing(b) Square footing
D
D
B/2 B/2
B/2B/2
B/2
D
Footing
Footing
Footing
Soil
Soil
Soil
P1P2
P3
P4P5
P1
P2
P3
P4
P51
2
3
1
2
3
1
2
u1 � u2 � 0
u1 � 0
u2 � 0u1 � 0
u1 � 0
Footing�has�u1f � 0,� u2f � � �,� u3f � 0
u1 � u2 � u3 � ur1 � ur2 � ur3 � 0u1 � u2 � u3 � ur1 � ur2 � ur3 � 0
P6
P7
vertical displacement only
P6
Interface�of��footing�sides� and� soil�P6�just�has
Elements�in�planes�P2 and P7�have�u1 � ur2 � ur3 � 0
Elements�in�planes�P1 and P6�have�u3 � ur1 � ur2 � 0
Elements�in�planes�P3, P4 and P5�have :
Elements�in�planes�P1�have�u3 � ur1 � ur2 � 0
Elements�in�planes�P2�have�u1 � ur2 � ur3 � 0
Elements�in�planes�P3, P4 and P5�have :
Footing�has�u1f � 0,� u2f � � �,� u3f � 0
77Chapter 4
P2
Footing Footing
FootingFooting
DD
DD
B/2B/2
L/2L/2
B/2 L/2
L/2B/2
Figure 4.4 Typical boundary conditions for rectangular footings
(a) Rectangular footing, B/L=0.50
(d) Rectangular footing, B/L=0.20(c) Rectangular footing, B/L=0.25
(b) Rectangular footing, B/L=0.33
P2P1
P5
P3
P4
2
31
1
2
3
P1
P3P4
P5
P2 P1
P3
P4
P5
P2P1
P3
P4
P5
2
31
1
2
3
SoilSoil
SoilSoil
P6P7
P7P6
P7
P6
P7
P6
Elements�in�planes�P2 and P7�have�u1 � ur2 � ur3 � 0
Elements�in�planes�P1 and P6�have�u3 � ur1 � ur2 � 0
Elements�in�planes�P3, P4 and P5�have :
Footing�has�u1f � 0,� u2f � � �,� u3f � 0
u1 � u2 � u3 � ur1 � ur2 � ur3 � 0
78Chapter 4
1
2
(a) Strip footing
B/2
1
2
3 (b) Square footing
Figure 4.5 Typical meshing of the finite element models for strip, square,
and circular footings
1
2
3
(c) Circular footing
79Chapter 4
1
2
3
D
Figure 4.6 Typical meshing of finite element models of rectangular footings
(a) Rectangular footing, B/L=0.50
(d) Rectangular footing, B/L=0.20(c) Rectangular footing, B/L=0.25
(b) Rectangular footing, B/L=0.33
1
2
3(a)
1
2
3(b)
1
2
3
(c) (d)
80Chapter 4
Figure 4.7 Element types used in the ABAQUS FE models.
(a) 8−node plane strain quadrilateral, hybrid, linear pressure, reducedintegration element for strip footing problems (CPE8RH)
b) 10−node modified quadratic tetrahedron element for square,circular and rectangular footing problems (C3D10M)
The overall mesh dimensions were selected to ensure that the zones of plastic shearing and
the observed displacement fields were contained within the model boundaries at all times.
The underside of all footings was modelled as perfectly rough by specifying a “tied”
contact constraint at the footing/soil interface.
To determine the collapse load of the footing, displacement defined analyses were
performed where the footing was considered as being perfectly rigid. That is, a uniform
vertical prescribed displacement (� �) was applied to all those nodes on the footing. The
total displacement was applied over a number of substeps and the nodal contact forces
along the footing were summed to compute the equivalent bearing capacity. The number
81Chapter 4
of displacement increments is automatically determined by ABAQUS, within initial,
minimum, and maximum values prescribed by users.
4.4 ABAQUS FINITE ELEMENT ANALYSIS RESULTS
In this section the numerical results obtained for the bearing capacity of embedded footings
in undrained clay are presented. These results are compared with results of previous
numerical and laboratory investigations.
The load−displacement behaviour of embedded footings is presented in Figure 4.8 to
Figure 4.11 in terms of the dimensionless parameters �Eu�Bcu, where � is the
displacement of the footing.
The plateaus shown in Figure 4.8 to Figure 4.11 indicate that collapse is clearly defined for
each embedment ratio D/B and footing shape.
For footings at relatively small embedment ratios (D/B�0.6) collapse is very clearly
defined because the load−displacement plot reaches a plateau at low displacements. The
failure mechanism zone can be seen in displacement vectors and contours in Figure 4.22
to Figure 4.25. For deeper footings (D/B>0.6) the displacements at collapse are very large.
For example, referring to Figure 4.9 for load−displacement behaviour of embedded square
footings, the displacements � at the collapse vary from 0.3B for D/B=0.2 to 1.5B for
D/B=2.
The plateaus shown in Figure 4.8 to Figure 4.11 also indicate that, for the same footing
depth at collapse, the value of the displacements of footing �Eu�Bcu increases as the
footing shape changes from strip, circular, square, rectangular B/L=0.5, B/L=0.33,
B/L=0.25 and B/L=0.2. For example, for D/B�0.6, at the collapse, �Eu�Bcu=65.5, 67.32,
80.23, 95.44, 104.09 and 132.12 as footing shape changes from strip, circular, square,
rectangular B/L=0.5, B/L=0.33, L=0.25 and B/L=0.2.
A summary of all the ABAQUS results is presented in Table 4.1 and Table 4.2. Also shown
in this table are the result of lower and upper bound limit analysis of Salgado et al 2004
and those of Skempton (1951) as shown in equation (2.15).
Chapter 4
82
Strip footing Square footing Circular footingD/B AB LB UB SK AB LB UB SK AB LB UB SK
0 5.24 5.132 5.203 5.14 5.95 5.523 6.227 6.168 6.048 5.856 6.227 6.1680.01 5.25 5.164 5.259 5.258 6.211 5.61 6.503 6.31 6.265 5.962 6.503 6.310.05 5.48 5.293 5.384 5.405 6.659 5.886 6.84 6.486 6.851 6.295 6.84 6.4860.1 5.67 5.448 5.548 5.514 7.063 6.171 7.14 6.617 7.324 6.491 7.14 6.6170.2 5.86 5.696 5.806 5.669 7.539 6.59 7.523 6.803 7.843 6.897 7.523 6.8030.4 6.27 6.029 6.133 5.888 8.123 7.194 8.104 7.066 8.334 7.303 8.104 7.0660.6 6.53 6.24 6.341 6.057 8.548 7.671 8.608 7.268 8.774 7.866 8.608 7.2680.8 6.771 6.411 6.509 6.198 8.921 8.068 9.034 7.438 9.183 8.37 9.034 7.438
1 6.871 6.526 6.657 6.323 9.271 8.429 9.429 7.588 9.574 8.771 9.429 7.5882 7.52 7.13 7.227 6.813 10.607 9.752 11.008 8.176 11.001 9.973 11.008 8.1763 7.931 7.547 7.652 7.19 12.108 10.532 12.14 8.627 12.081 10.686 12.14 8.6274 8.166 7.885 7.994 7.5 13.075 10.941 13.03 9 12.978 10.954 13.03 95 8.498 8.168 8.284 7.5 13.848 11.206 13.743 9 13.7 10.998 13.743 9
Note: AB is results of ABAQUS in this study
LB is lower bound and UB is upper bound result of Salgado et al (2004)
SK is the results of Skempton (1951) in equation (2.15)
Table 4.1 Normalised undrained bearing capacity for embedded strip, square and circular footings (qu�cu), compared to result of Salgado et
al (2004) and Skempton (1951) in equation (2.15)
Chapter 4
83
Rectangular footing, B/L=0.5 Rectangular footing, B/L=0.33 Rectangular footing, B/L=0.25 Rectangular footing, B/L=0.2D/B AB LB UB SK AB LB UB SK AB LB UB SK AB LB UB SK
0 5.73 5.359 6.022 5.654 5.613 5.256 5.886 5.479 5.54 5.201 5.82 5.397 5.513 5.169 5.776 5.3460.01 5.822 5.424 6.249 5.784 5.669 5.311 6.085 5.605 5.572 5.253 6.006 5.521 5.572 5.218 5.949 5.4690.05 6.186 5.64 6.503 5.945 5.983 5.503 6.3 5.761 5.92 5.43 6.203 5.675 5.839 5.389 6.126 5.6210.1 6.596 5.86 6.756 6.066 6.154 5.697 6.533 5.878 6.122 5.614 6.413 5.79 6.136 5.565 6.321 5.7350.2 6.947 6.197 7.116 6.236 6.63 5.997 6.867 6.043 6.523 5.895 6.731 5.953 6.529 5.836 6.637 5.8960.4 7.448 6.68 7.574 6.477 7.112 6.408 7.271 6.277 6.965 6.272 7.113 6.183 6.95 6.19 7.003 6.1240.6 7.835 7.082 7.993 6.662 7.417 6.74 7.608 6.456 7.239 6.567 7.412 6.359 7.209 6.465 7.299 6.2990.8 8.185 7.427 8.377 6.818 7.709 7.03 7.936 6.607 7.485 6.717 7.705 6.508 7.421 6.695 7.57 6.446
1 8.482 7.729 8.724 6.956 7.994 7.297 8.24 6.741 7.726 7.048 7.976 6.639 7.619 6.904 7.819 6.5762 9.692 8.968 10.055 7.495 9.131 8.447 9.476 7.263 8.719 8.109 9.086 7.154 8.525 7.86 8.835 7.0863 10.593 9.86 11.076 7.909 9.987 9.296 10.473 7.664 9.545 8.92 10.026 7.549 9.309 8.607 9.696 7.4774 11.38 10.513 11.878 8.257 10.698 10.018 11.242 8.002 10.259 9.594 10.769 7.882 9.961 9.249 10.403 7.8075 12.043 10.88 12.545 8.565 11.347 10.464 11.887 8.3 10.894 10.117 11.408 8.175 10.581 9.796 11.03 8.097
Note: AB is results of ABAQUS in this study
LB is lower bound and UB is upper bound result of Salgado et al (2004)
SK is the results of Skempton (1951) in equation (2.15)
Table 4.2 Normalised undrained bearing capacity for embedded rectangular footings (qu�cu), compared to result of Salgado et al (2004) and
Skempton (1951) in equation (2.15)
Chapter 4
84
0
1
2
3
4
5
6
7
0 20 40 60 80 100 120 140
Figure 4.8 Load−displacement behaviour of embedded strip footing for
�Eu�Bcu
Qu
Bcu
D�B � 0.6
D�B � 0.4
D�B � 0.2
D�B � 0 D�B � 0.01D�B � 0.05D�B � 0.1
0 � D�B � 0.6
Chapter 4
85
0
2
4
6
8
0 50 100 150 200 250 300 350 400 450 500
Figure 4.8 (continued) Load−displacement behaviour of embedded strip footing
�Eu�Bcu
Qu
Bcu
D�B � 5D�B � 4
D�B � 3
D�B � 0.8D�B � 1
D�B � 2
for 0.8 � D�B � 5
Chapter 4
86
0
2
4
6
8
0 20 40 60 80 100 120
Figure 4.9 Load−displacement behaviour of embedded square footing
�Eu�Bcu
Qu
Bcu
D�B � 0.6
D�B � 0.4
D�B � 0.2
D�B � 0
D�B � 0.01
D�B � 0.05
D�B � 0.1
for 0 � D�B � 0.6
Chapter 4
87
0
2
4
6
8
10
12
14
0 30 60 90 120 150 180 210
Figure 4.9 (continued) Load−displacement behaviour of embedded square footing
�Eu�Bcu
Qu
Acu
D�B � 5
D�B � 4
D�B � 3
D�B � 0.8
D�B � 1
D�B � 2
for 0.8 � D�B � 5
Chapter 4
88
0
2
4
6
8
0 30 60 90 120 150 180 210 240 270 300
Figure 4.10 Load−displacement behaviour of embedded circular footing
�Eu�Bcu
Qu
Acu
D�B � 0.8
D�B � 0.4
D�B � 0.2
D�B � 0
D�B � 0.01
D�B � 0.05D�B � 0.1
D�B � 0.6
for 0 � D�B � 0.8
Chapter 4
89
0
2
4
6
8
10
12
14
0 40 80 120 160 200 240 280 320 360
Figure 4.10 (continued) Load−displacement behaviour of embedded circular footing
�Eu�Bcu
Qu
Acu
D�B � 5
D�B � 4
D�B � 3
D�B � 1
D�B � 2
for 1 � D�B � 5
Chapter 4
90
0
1
2
3
4
5
6
7
8
0 20 40 60 80 100 120 140
Figure 4.11 Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 0.4
D�B � 0.2
D�B � 0D�B � 0.01
D�B � 0.05 D�B � 0.1
for B�L � 0.5 and� 0 � D�B � 0.4
Chapter 4
91
0
2
4
6
8
10
12
0 60 120 180 240 300 360 420 480 540 600
Figure 4.11 (continued) Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 5
D�B � 4
D�B � 3
D�B � 1
D�B � 2
D�B � 0.6
D�B � 0.8
for B�L � 0.5 and 0.6 � D�B � 5
Chapter 4
92
0
1
2
3
4
5
6
7
8
0 20 40 60 80 100 120
Figure 4.11 (continued) Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 0.6
D�B � 0.2
D�B � 0D�B � 0.01D�B � 0.05
D�B � 0.1
D�B � 0.4
for B�L � 0.33 and 0 � D�B � 0.6
Chapter 4
93
0
2
4
6
8
10
12
0 50 100 150 200 250 300 350 400 450
Figure 4.11 (continued) Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 5
D�B � 4
D�B � 3
D�B � 1
D�B � 2
D�B � 0.8
for B�L � 0.33 and 0.8 � D�B � 5
Chapter 4
94
0
1
2
3
4
5
6
7
0 20 40 60 80 100 120
Figure 4.11 (continued) Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 0.6
D�B � 0.2
D�B � 0D�B � 0.01D�B � 0.05D�B � 0.1
D�B � 0.4
for B�L � 0.25 and 0 � D�B � 0.6
Chapter 4
95
0
2
4
6
8
10
0 60 120 180 240 300 360 420 480 540 600
Figure 4.11 (continued) Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 5
D�B � 4
D�B � 3
D�B � 1
D�B � 2
D�B � 0.8
for B�L � 0.25 and 0.8 � D�B � 5
Chapter 4
96
0
1
2
3
4
5
6
7
0 30 60 90 120 150 180 210
Figure 4.11 (continued) Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 0.2
D�B � 0D�B � 0.01D�B � 0.05
D�B � 0.1
D�B � 0.4
for B�L � 0.2 and 0 � D�B � 0.4
Chapter 4
97
0
2
4
6
8
10
0 60 120 180 240 300 360 420 480 540 600
Figure 4.11 (continued) Load−displacement behaviour of embedded rectangular footing
�Eu�Bcu
Qu
Acu
D�B � 5D�B � 4
D�B � 3
D�B � 1
D�B � 2
D�B � 0.8D�B � 0.6
for B�L � 0.2 and 0.6 � D�B � 5
98Chapter 4
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5qu�cu
DB
Figure 4.12 Normalised bearing capacity of embedded strip footings
UB−Salgado et al (2004)
LB−Salgado et al (2004)
Skempton (1951) (equation 2.15)
FEM − ABAQUS
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
4.3 5.3 6.3 7.3 8.3 9.3 10.3 11.3 12.3 13.3qu�cu
DB
Figure 4.13 Normalised bearing capacity of embedded square footings
UB−Salgado et al (2004)
LB−Salgado et al (2004)
Skempton (1951) (equation 2.15)
FEM − ABAQUS
99Chapter 4
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
4.3 5.3 6.3 7.3 8.3 9.3 10.3 11.3 12.3 13.3qu�cu
DB
Figure 4.14 Normalised bearing capacity of embedded circular footings
UB−Salgado et al (2004)
LB−Salgado et al (2004)
Skempton (1951) (equation 2.15)
FEM − ABAQUS
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5qu�cu
DB
Figure 4.15 Normalised bearing capacity of embeddedrectangular footing for B/L=0.5
UB−Salgado et al (2004)
LB−Salgado et al (2004)
Skempton (1951) (equation 2.15)
FEM − ABAQUS
100Chapter 4
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5qu�cu
DB
Figure 4.16 Normalised bearing capacity of embeddedrectangular footing for B/L=0.33
UB−Salgado et al (2004)
LB−Salgado et al (2004)
Skempton (1951) (equation 2.15)
FEM − ABAQUS
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5qu�cu
DB
Figure 4.17 Normalised bearing capacity of embeddedrectangular footing for B/L=0.25
UB−Salgado et al (2004)
LB−Salgado et al (2004)
Skempton (1951) (equation 2.15)
FEM − ABAQUS
101Chapter 4
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
4.5 5.5 6.5 7.5 8.5 9.5 10.5qu�cu
DB
Figure 4.18 Normalised bearing capacity of embeddedrectangular footing for B/L=0.20
UB−Salgado et al (2004)LB−Salgado et al (2004)
Skempton (1951) (equation 2.15)
FEM − ABAQUS
−6.0
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5qu�cu
Figure 4.19 Normalised bearing capacity of embedded footings
DB
Square
Circlular footing
Square footing
Strip footing Rectangular footing B/L=0.50
Rectangular footing B/L=0.20
Rectangular footing B/L=0.25
Rectangular footing B/L=0.33
StripBL� 0.2
Circular
BL� 0.25 B
L� 0.33
BL� 0.5
BL� 1
BL� 1
BL� 0
102Chapter 4
It is clear seen from Table 4.1 and Table 4.2 that deeper foundations mobilise larger
volumes of soil and dissipate more plastic energy, therefore the larger bearing capacities
are at larger D�B ratios.
For the same value of D�B, the bearing capacity increases as the ratio of B/L rises from
0 (for strip footings) to 1 (for square or circular footings). At the same depth , if D�B � 3,
the bearing capacity of circular footings is larger than of square footings by about 2−4%
and if D�B � 3 bearing capacity of circular footings is smaller than those of square
footings by about 1−3%.
The equation (4.1) assumes implicitly that shape factors Fcs and depth factors Fcd are
independent from each other. The shape factors will be a equation of B�L and be calculated
by bearing capacity of surface footings ( D�B � 0) for strip, square, circular and
rectangular footings. These shape factors do not depend on the footing depths.
The depth factors Fcd are equations of both parameters B�L and D�B and calculated from
(4.1) when the shape factors have been determined beforehand.
4.4.1 Shape factors
Shape factors Fcs can be calculated using the values of undrained bearing capacity for
embedded footings qu�cu in Table 4.1 and Table 4.2 in conjunction with equation (4.1).
The shape factor Fcs is determined by dividing the bearing capacity factor Nc for any
surface footing by Nc for strip surface footing.
− For strip footings, shape factor Fcs=1.
− For a square footing, shape factor is determined as:
Fcs �N[square�surface�footing]
c
N�[strip�surface�footing]c
� 5.955.24
� 1.136 (4.2)
where N[square�surface�footing]c is the bearing capacity factor of a square surface footing,
from Table 4.1, N[square�surface�footing]c = 5.95.
N[strip�surface�footing]c is the bearing capacity factor of a surface strip footing, from
Table 4.1, N[strip�surface�footing]c = 5.24.
103Chapter 4
− For a circular footing, shape factor is determined as:
Fcs �N[circular�surface�footing]
c
N�[strip�surface�footing]c
� 6.055.24
� 1.155 (4.3)
where N[circular�surface�footing]c is the bearing capacity factor of a circular surface footing,
N[circular�surface�footing]c = 6.05 (Table 4.1).
N[strip�surface�footing]c is the bearing capacity factor of a surface strip footing,
N[strip�surface�footing]c = 5.24 (Table 4.1).
− For a rectangular footing, shape factor is determined as:
Fcs �N[rectangular�surface�footing]
c
N�[strip�surface�footing]c
(4.4)
where N[rectangular�surface�footing]c is the bearing capacity factor of a circular surface
footing, N[rectangular�surface�footing]c (Table 4.2).
N[strip�surface�footing]c is the bearing capacity factor of a surface strip footing,
N[strip�surface�footing]c = 5.24 (Table 4.1).
Fcs � 5.735.24
� 1.094 for rectangular footing B�L � 0.5;
Fcs � 5.6135.24
� 1.071 for rectangular footing B�L � 0.33;
Fcs � 5.545.24
� 1.057 for rectangular footing B�L � 0.25;
Fcs � 5.5135.24
� 1.052 for rectangular footing B�L � 0.2;
All of these values of shape factors Fcs are summarized in Table 4.3.
104Chapter 4
Footing types B�L Fcs
Circular footing 1 1.155
Square footing 1 1.136
Rectangular
footing
0.50 1.094
0.33 1.071
0.25 1.057
0.20 1.052
Strip footing 0 1.000
Table 4.3 Shape factors for various footing geometries
For surface footings the shape factor Fcs can be extracted as a function of the ratio of footing
dimensions B�L. In the present work, B�L =0.5, 0.33, 0.25 and 0.2 which covers most
of the practical cases of rectangular footings, B�L=0 for strip footings and B�L=1 for
square and circular footings.
The shape factor Fcs increases as the ratio of relative dimensions of footing B�L rises from
0 (for strip footing) to 1 (for square and circular footing). However, when B�L � 0.2, the
shape factor Fcs increases rapidly and when B�L � 0.2, this increase is slower as the
ration B�L increases. The shape factor Fcs of circular footing is higher than that of square
footing.
For 0 � B�L � 1, the values of the shape factor Fcs in Table 4.3 can be approximated by
the following equation:
Fcs � 1 � 0.136BL0.56
for 0 � B�L � 1 (4.5)
Equation (4.5) provides estimates of the shape factors to within � 0.5% of the finite
element solutions (ABAQUS) and therefore can be used with confidence to solve practical
design problems.
The shape factors of ABAQUS in Table 4.3, estimated equation (4.5), and equation (2.13)
of Meyerhof (1951) are presented in Figure 4.20.
105Chapter 4
Figure 4.20 Comparison of the shape factors
0.95
1.00
1.05
1.10
1.15
1.20
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Fcs
B�L
Fcs
Estimated by equation (4.5)
ABAQUS
Meyerhof (1951)
In the equation (4.1), the shape factor Fcs is assumed to be a function of B�L only and not
the embedment depth D�B. This shape factor Fcs will be used when calculating the depth
factor Fcd. Therefore equation (4.5) can be used to determine shape factors Fcs for both
surface footings and embedded footings.
4.4.2 Bearing capacity and depth factor for strip footings
The bearing capacity factors Nc from the current finite element study compare well to that
calculated using the widely adopted depth factor in the empirical equation (2.15) of
Skempton (1951), and the numerical solutions of Salgado et al (2004). They all increase
gradually as value of D�B rises. The results of this study were found to be around 4.2 and
6.4% above that for the upper and lower bound work of Salgado et al (2004), and around
11% above the Skempton (1951) solutions. For example, at the same value of D�B=3.0,
the value of bearing capacity factors Nc from ABAQUS, upper bound, lower bound and
Skempton solutions are 7.931, 7.652, 7.547 and 7,19 respectively.
106Chapter 4
In the solutions of Skempton (1951), the bearing capacity factor Nc reaches a maximum
value of 7.5 when D�B � 4.0 whereas in this study and in the upper and lower bound limit
analysis of Salgado et al (2004), the bearing capacity factor Nc does not reach a limiting
value.
For embedment depths of D�B � 0.6, the modes of failure can best be described as being
general shear, as illustrated in Figure 4.22 and Figure 4.23. As can be seen from
Figure 4.22 and Figure 4.23, the modes of failure are similar to those of surface footings
on undrained soil. The soil elements with a greatest displacement are underneath and along
the centre line of the footing. When D�B � 0.6 (Figure 4.23(d)−(i))), the failure
mechanism is similar to that of a local shear failure of a surface footing. The failure zone
is deeper and is located more in the vicinity of the footing as the ratio D�B increases. For
values of D�B � 0.6, more soil elements move horizontally away from the footing instead
of moving to the surface, see Figure 4.23(h) and Figure 4.23(i). It is likely that when the
ratio D�B rises to a infinite depth, soil displacements will no longer be observed at the
surface. In that case, the shear failure mechanism will be contained around the footing toe.
The depth factor Fcd of a strip footing can be calculated using values in Table 4.1 in
conjunction with equation (4.1). These depth factors Fcd are shown in Table 4.4. Fcd
increases gradually with the ratio of D�B. In this study, the depth factors Fcd are
determined as equations of two variables of D�B and B�L and calculated by dividing the
bearing capacity factor Nc at various D�B values by Nc for surface footing. For strip
footing, B�L=0, the depth factors Fcd depend on D�B only. The depth factors Fcd results
of Salgado et al (2004) (Fcd � 1 � 0.27 DB
), and of Skempton (1951)
Fc � �1 � 0.053 DB �, and of ABAQUS are shown in Figure 4.31.
107Chapter 4
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65
Figure 4.21 Depth factors for strip footing as calculated by ABAQUS andexpressions proposed by Salgado et al (2004), and by Skempton (1951)
DB
Fcd
From Salgado et al (2004) as equationFcd � 1 � 0.27(D�B)0.5
From Skempton (1951) as equation Fcd � 1 � (0.053D�B)0.5
From FEM−ABAQUS
As can be seen from Figure 4.21, at the same embedment depths D�B, depth factor of this
study is very close to that of Salgado et al (2004) equation (higher than 1−3%) but much
higher than result of Skempton (1951) (maximum at 28%). That differences are larger as
footing depths are deeper.
4.4.2 Bearing capacity and depth factor for square footings
Figure 4.13 and Table 4.1 present the undrained bearing capacity factors of embedded
square footings. The results of the present study based on the finite element method are very
close to those of upper bound limit analysis solutions of Salgado et al. (2004). The
numerical solutions differ from each other by only around 0.5% while the empirical
solutions of Skempton (1951) are very low relative to both numerical solutions. For
example, at D�B � 5.0, values of Nc in the present and the previous upper and lower
bound limit analysis are 13.848, 13.640 and 11.206 respectively and Nc is still increasing
108Chapter 4
with D�B at this point whereas the Skempton (1951) solution for Nc reaches a constant
value of 9.0 by this D�B ratio.
The displacement vectors and contours for embedded square footings are shown in
Figure 4.24. When the embedment depth of footings is small, D�B � 0.5, the failure
mechanism is very similar to that for surface footings. The passive zones outside the footing
have slip surfaces as in general shear failure, but the displacement vectors are at about 45°
with respect to the horizontal surface and rise to almost 90° when the ratio D�B increases
to 0.5. When 0 � D�B � 0.5, the bearing capacity increase rapidly, but when
0.2 � D�B � 5 there is a more gradual increase in bearing capacity. This can be seen
clearly in Figure 4.13. When the ratio D�B rises above 0.5, the zone of failure underneath
the footing becomes more localised near the footing. Some soil goes around to the side of
the footing and pushes a large number of soil elements upward.
In this study, for embedded square footings, depth factors Fcd are calculated by dividing
the bearing capacity factors Nc of embedded square footing at the various D�B by Nc of
of surface square footing. The depth factors Fcd for embedment square footing calculated
from results of this study are presented in Table 4.4 and shown in Figure 4.27.
For embedment square, circular and rectangular footings, the determination of depth
factors Fcd of this study is different from that of results of Skempton (1951) and Salgado
et al. (2004). The depth factors Fcd of results of Skempton (1951) and Salgado et al. (2004)
are the same for any shape footing, while depth factors Fcd of this study change from
footing shape to footing shape.
4.4.3 Bearing capacity and depth factor for circular footings
The bearing capacity factor Nc of this study was found to be almost identical for square
and circular footings for all cases. In the cases of D�B � 3.0, the bearing capacity factor
for square footings was around 1−2% above that of a circular footing. But in the cases of
D�B � 3.0 the bearing capacity factor for circular footings was around 0.5% above that
of a square footing. This is indicates some level of mesh dependency and is illustrated in
Figure 4.19.
The modes of failures and displacement contours can be seen in Figure 4.25, and are similar
to embedded square footings as discussed in Section 4.4.2. These results are very close to
109Chapter 4
the upper bound limit analysis of Salgado et al. (2004), but larger than the results of
Skempton (1951), see Figure 4.14. When the ratio D�B increases from 0 to 0.2, the failure
mechanism is similar to that of a footing on the surface of the same soil medium. Within
this range of D�B values, the normalised undrained bearing capacity of embedded circular
footings increases rapidly from 6.048 to 7.843 (Table 4.1). However, for D�B � 0.20, the
normalised undrained bearing capacity rises gradually to 13.70 when D�B � 5.
In this study, for embedded circular footings, depth factors Fcd are also calculated by
dividing the bearing capacity factors Nc of embedded circular footings at the various D�B
by Nc of the surface circular footing. The depth factors Fcd for embedment circular footing
calculated from results of this study are presented in Table 4.4 and shown in Figure 4.27.
4.4.4 Bearing capacity and depth factor for rectangular
footings
Results of this study on the bearing capacity of embedded rectangular footings are
presented in Table 4.2 and graphically in the Figure 4.26.
In general, the results of this study are similar to those obtained from the limit analysis of
Salgado et al (2004), tending to be closer to the upper bound than the lower bound results.
Bearing capacities of embedded rectangular footing increase as ratios D�B and B�L
increase.
In the surface footing cases, the normalised bearing capacities of all the rectangular
footings from the finite element analysis of this study and from the laboratory
investigations of Skempton (1951) are very close to each other (within about 1−2%),
Table 4.2. This suggests the shape factor Fcs suggested by Skempton (1951) is similar to
the limit analysis results. Again, for embedded footings the bearing capacity factors
proposed by Skempton (1951) appear very conservative.
From Figure 4.19, the value of normalised bearing capacity factors Nc for rectangular
footings increase as ratio B�L rises. For example, at the depth of D�B � 3, the
normalised bearing capacity factor Nc for rectangular footing of B�L=0.2, 0.25, 0.33 and
0.50 are 9.309, 9.545, 9.987 and 10.593 respectively.
The typical displacement vectors and contours for rectangular footing problems in the
symmetry planes P1 and P2 are presented in Figure 4.26. The failure mechanism in the
110Chapter 4
planes P1 is larger and increasingly happens in the vicinity of the footing as the ratio of
B�L increases from 0.2 to 0.5 for cases where D�B � 4. However, for cases where
D�B � 4, failure tends to be localised around the footing instead of extending to the
surface. In the plane P2, when B�L increases from 0.2 to 0.5 the displacement vectors and
contours change the active zone from isosceles trapezium to isosceles triangle . This is
consistent with the case of strip footings (see Figure 4.23(d) − (i)) which corresponds to
a rectangular footing in the case where B�L � 0.
In this study, for embedded rectangular footings, depth factors Fcd are also calculated by
dividing the bearing capacity factors Nc of embedded rectangular footings at the various
D�B by Nc of the surface rectangular footing. The depth factors Fcd for embedment
rectangular footings calculated from results of this study are presented in Table 4.4 and
shown in Figure 4.27.
The depth factors Fcd of all footings shapes in this study are determined differently from
those of Skempton (1951) and Salgado et al. (2004), which are presented in Table 4.4 and
shown in Figure 4.27. The depth factors Fcd contained both parameters D�B and B�L. For
practical uses, the depth factors Fcd in Table 4.4 and Figure 4.27 can be approximated by
following equation of two parameters D�B and B�L.
Fcd � 1 � 0.48 � 0.4e�B�L DB
(4.6)
Equation (4.6) provides estimates of the depth factors to within � 3.5% of the finite
element solutions (ABAQUS) and therefore can be used with confidence to solve practical
design problems. Figure 4.28 presents graphically the depth factors from ABAQUS and
from the estimated equation (4.6).
111Chapter 4
Figure 4.22 Displacement contours for strip footings D/B= 0.01, 0.05 and 0.1
d) D/B=0.1
b) D/B=0.01
c) D/B=0.05
112Chapter 4
1
Figure 4.23 Displacement vectors and contours for strip footings D/B=0.2, 0.4 and 0.6
a) D/B=0.2
b) D/B=0.4
c) D/B=0.6
113Chapter 4
Figure 4.23 (continued) Displacement vectors and contours for strip footings D/B=0.8, 1.0 & 2.0f) D/B=2.0
e) D/B=1.0
d) D/B=0.8
114Chapter 4
Figure 4.23 (continued) Displacement vectors and contours for strip footings D/B=3.0, 4.0 & 5.0
i) D/B=5.0
h) D/B=4.0
g) D/B=3.0
115Chapter 4
Figure 4.24 Displacement vectors and contours for square footings D/B=0.01, 0.05 & 0.1
a) D/B=0.01
b) D/B=0.05
c) D/B=0.1
116Chapter 4
Figure 4.24 (continued) Displacement vectors and contours for square footings D/B=0.2, 0.4 & 0.6
d) D/B=0.2
e) D/B=0.4
f) D/B=0.6
117Chapter 4
Figure 4.24 (continued) Displacement vectors and contours for square footings D/B=0.8, 1.0 & 2.0
i) D/B=2.0
h) D/B=1.0
g) D/B=0.8
118Chapter 4
Figure 4.24 (continued) Displacement vectors and contours for square footings D/B=3.0, 4.0 and 5.0
k) D/B=4.0
j) D/B=3.0
l) D/B=5.0
119Chapter 4
Figure 4.25 Displacement vectors and contours for circular footings D/B=0.01, 0.1 and 0.2
b) D/B=0.10
a) D/B=0.01
c) D/B=0.2
120Chapter 4
Figure 4.25 (continued) Displacement vectors and contours for circular footings D/B=0.4, 0.6 and 0.8
e) D/B=0.6
d) D/B=0.4
f) D/B=0.8
121Chapter 4
Figure 4.25 (continued) Displacement vectors and contours for circular footings D/B=1.0, 3.0 & 5.0
i) D/B=5.0
h) D/B=3.0
g) D/B=1.0
122Chapter 4
Figure 4.26 Displacement vectors and contours for rectangular footings L/B=0.50, D/B=0.4
1
2
3
1
2
3
a)�Plane�P1
b)�Plane�P2
123Chapter 4
1
2
3
1
2
3
Figure 4.26 (continued) Displacement vectors and contours for rectangular footings L/B=0.50, D/B=4.0
c)�Plane�P1
d)�Plane�P2
124Chapter 4
Figure 4.26 (continued) Displacement vectors and contours for rectangular footings L/B=0.33, D/B=0.4
1
2
3
12
3
e)�Plane�P1
f)�Plane�P2
125Chapter 4
Figure 4.26 (continued) Displacement vectors and contours for rectangular footings L/B=0.33, D/B=4.0
1
2
3
1
2
3
g)�Plane�P1
h)�Plane�P2
126Chapter 4
Figure 4.26 (continued) Displacement vectors and contours for rectangular footings L/B=0.20, D/B=0.4
1
2
3
1
2
3
i)�Plane�P1
j)�Plane�P2
127Chapter 4
Figure 4.26 (continued) Displacement vectors and contours for rectangular footings L/B=0.2, D/B=4.0
1
2
3
1
2
3
k)�Plane�P1
l)�Plane�P2
128Chapter 4
D/BStripfooting,
Squarefooting,
Circularfooting,
Rectangular footing
B/L=0 B/L=1 B/L=2 B/L=0.50 B/L=0.33 B/L=0.25 B/L=0.200 1 1 1 1 1 1 1
0.01 1.002 1.044 1.036 1.016 1.01 1.006 1.0110.05 1.046 1.119 1.133 1.08 1.066 1.069 1.0590.1 1.082 1.187 1.211 1.151 1.096 1.105 1.1130.2 1.118 1.267 1.297 1.212 1.181 1.177 1.1840.4 1.197 1.365 1.378 1.3 1.267 1.257 1.2610.6 1.246 1.437 1.451 1.367 1.321 1.307 1.3080.8 1.292 1.499 1.518 1.428 1.373 1.351 1.346
1 1.311 1.558 1.583 1.48 1.424 1.395 1.3822 1.435 1.783 1.819 1.691 1.627 1.574 1.5463 1.514 2.035 1.998 1.849 1.779 1.723 1.6894 1.558 2.197 2.146 1.986 1.906 1.852 1.8075 1.622 2.327 2.265 2.102 2.022 1.966 1.919
Table 4.4 Depth factors Fcd for embedded footings from ABAQUS
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Fcd
DB
Square
Circular footing
Square footing
Strip footing Rectangular footing B/L=0.50
Rectangular footing B/L=0.20
Rectangular footing B/L=0.25
Rectangular footing B/L=0.33
B�L � 0.2
Circular
B�L � 0.25 B�L � 0.33 B�L � 0.5
BL� 1
BL� 1
B�L � 0
Figure 4.27 Depth factors for embedded footings from ABAQUS
Strip
129Chapter 4
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Fcd
DB
Circlular footing
Square footing
Strip footing Rectangular footing B/L=0.50
Rectangular footing B/L=0.20
Rectangular footing B/L=0.25
Rectangular footing B/L=0.33
B�L � 0.2
Circular, Square
B�L � 0.25 B�L � 0.33 B�L � 0.5
BL� 1
B�L � 0
Figure 4.28 Depth factors for embedded footings from ABAQUS and equation (4.6)
Strip
Note that: The marked points are ABAQUS results,
the curves are the estimated equation (4.6)
4.5 CONCLUSIONS
The ultimate undrained bearing capacity of strip, square, circular and rectangular
embedded footings has been investigated by finite element method. The soil is treated as
weightless. The interactions between soil and the bottom of the footings are treated as being
fully rough. The results obtained have been presented in terms of a normalised bearing
capacity factors Nc in both graphical and tabular form; and the shape factors Fcs; and the
depth factors Fcd to facilitate their use in solving practical design problems.
The following conclusions can be made based on the finite element results:
130Chapter 4
� In general, the finite element undrained bearing capacity of embedded footing
compares well to the previously reported numerical limit analysis solutions. The finite
element results are higher than that of Skempton (1951).
� For strip footings, the bearing capacity factors are higher than that of both the upper
and lower bound solutions of Salgado et al (2004). This is somewhat unexpected and
may suggest a certain level of mesh dependency in the FE results. For square and
circular footing cases, the results are very close to the upper bound limit analysis, while
for rectangular footings, the bearing capacity factors lie between upper and lower
bound solutions.
� For cases where D�B � 0.2 for strip footing, D�B � 0.4 for rectangular footing,
and D�B � 0.1 for square and rectangular footings the failure mechanism is similar
to that of a surface footing. However when the D�B is larger, the failure mechanism
is localised near the footing.
� Bearing capacity continues to increase for D�B � 4, and this is different from the
conclusion of Skempton (1951) where the bearing capacity factor reaches a constant
value in the case of D�B � 4. The inclusion of soil weight is likely to limit the growth
in bearing capacity indefinitely.
� The present work also gives the shape and depth factors in the form of an equation and
tables to aid in their use.
131Chapter 5
CHAPTER 5
UNDRAINED BEARING CAPACITY OF
FOOTINGS NEAR SLOPES
132Chapter 5
5.1 INTRODUCTION
Shallow foundations are often placed on slopes, adjacent to slopes or near proposed
excavations.
In this chapter, the ultimate undrained bearing capacity of footings near slopes is
investigated. Consideration is given to the effect of slope angle, height of slope, the distance
from the edge of the slope to the footing, soil properties and the interaction between the
soil and footing. Estimates of the bearing capacity of footings on the surface of slopes are
obtained using the commercial software package ABAQUS.
There are two possible collapse modes for a footing located near a slope, namely due to
overall slope stability or bearing capacity failure. Both of these failure modes are affected
not only by the footing load but also by the soil body forces. For these reasons, the problem
of footings near slopes is more complex than that of surface footings on flat ground
(Chapter 3 and Chapter 4).
This chapter is primarily concerned with estimating the vertical bearing capacity of
footings on the surface of the slope in purely cohesive soil. Soil properties and the boundary
conditions are able to be modelled using the finite element method. The problem can be
solved using a two dimensional analysis because a state of plane strain has been assumed.
In this work, soil is homogeneous with unit weight and uniform undrained soil strength.
5.2 PROBLEM DEFINITION
A general layout of the problem to be analysed is shown in Figure 5.1. The system includes
a rigid footing and a clay soil slope. The bearing force Qu of a footing can be expressed
as:
Qu � quB (5.1)
where: B = footing width;
qu = the ultimate bearing capacity.
Because the soil unit weight � influences the overall stability of the slope, it will affect the
ultimate bearing capacity. For this problem, the ultimate bearing capacity qu can be
expressed in an equation (5.2) with dimensionless parameters:
133Chapter 5
qu
�B� f�°, L
B,
cu
�B, HB (5.2)
where: �°= slope angle;
L = distance from footing to the crest of the slope;
cu = the undrained soil strength;
� = the soil unit weight;
H = the slope height;
B = footing width.
The soil unit weight (i.e. gravity) and footing loads are the forces which cause instability,
while resistance to failure is derived from a combination of slope geometry and soil
cohesion.
The slope is assumed to be homogeneous clay (internal friction angle � � 0), isotropic
elasto−perfectly plastic with a failure described by the Mohr−Coulomb yield criterion. The
elastic behaviour was described by a Poisson’s ratio � � 0.49, shear strength cu, Young’s
modulus E, a ratio of Young’s modulus to shear strength of E�cu � 500, and soil unit
weight �.
The slope has a slope angle �°, and slope height H (Figure 5.1). H�B � 3 for the cases
of �°=15° and 30° (Figure 5.1a); H�B � 5 for the cases of �°=45°, 60°, 75° and 90°
(Figure 5.1b). The lower, left, and right limits of the problem domain are considered to be
sufficiently far away from the slope so as not to influence the solution. This was achieved
by first testing larger models where each side was 10 times the footing width, reductions
in dimensions was made while ensuring the results of the bearing capacity changed by less
than 2% from the solution obtained in the largest configuration.
In this work, the solutions have been computed for problems where:
* the slope angle �° varies between 15° and 90° in increments of 15°.
* the ratio cu��B ranges from 0 to 10.
* the ratio L�B varies between 0 and 9 in increments of 1.
These ranges cover most problems of practical interest.
The output of the computation will be the ultimate bearing capacity qu which can be
calculated using equation (5.2).
Chapter 5
134Figure 5.1 Problem definition for footings near slopes
ÉÉÉÉÉÉÉÉ
ÉÉÉÉ
Footing
Footing
�°
�°
L B
L BH � 3B
H � 5B
Homogeneous clay slope
Homogeneous clay slope
a) for� the�cases�of��° � 15° and 30°
b)� for� the�cases�of��° � 45°, 60°, 75° and 90°
DH
DH
Qu � quB
Qu � quB
135Chapter 5
5.3 FINITE ELEMENT MODELLING DETAILS
The finite element software ABAQUS was used for solving this problem. The ABAQUS
model consisted of two parts, namely the footing and the homogeneous clay slope. At the
interface between the footing and the soil, the elements representing the footing and those
representing the soil have the same topology.
Typical meshes for the problem of the two−dimensional strip footings on the surface of the
slope, including a magnified zone of meshing, along with the applied displacement
boundary conditions, are shown in Figure 5.2 and Figure 5.3. The mesh consisted of
8−node plane strain quadrilateral, hybrid, linear pressure, reduced integration element
(CPE8RH) as shown in Figure 5.4 because this element was found to provide the best
solution convergence. The mesh averages 7000 quadrilateral elements and 24000 nodes
depending on the positions of the footing and the slope angles.
The overall mesh dimensions were selected to ensure that the zones of plastic shearing and
the observed displacement fields were contained within the model boundaries at all times.
Typically, the lower boundary was fixed as it has no influence on the footing. The left−hand
and the right−hand edge of the domain was fixed vertically only, however the domain is
large enough so the footing also does not affect the left−hand and right−hand edge.
When a rough footing interaction was simulated, the underside of all footings was modelled
as perfectly rough by specifying a “tied” contact constraint at the footing/soil interface.
When a smooth footing interaction was simulated, the underside of all footings was
modelled as perfectly smooth by specifying a normal contact constraint at the footing/soil
interface.
To determine the collapse load of the footing, displacement defined analyses were
performed where the footing was considered as being perfectly rigid (i.e. a very large
Young’s Modulus was assumed). That is, a uniform vertical prescribed displacement was
applied to all those nodes on the footing. The total displacement was applied over a number
of sub steps and the nodal contact forces along the footing were summed to compute the
equivalent bearing capacity. The number of displacement increments is automatically
determined by ABAQUS, within initial, minimum, and maximum values prescribed by
users.
Chapter 5
136Figure 5.2
1
2
3
Footing
Level surface of slope
u1 � 0
ur2 � 0
u1 � 0
ur2 � 0
u1 � u2 � u3 � ur1 � ur2 � ur3 � 0
Slope surface of slope
Typical�meshing�of�finite�element�models�(for�the�cases�of��° � 15° and 30°)
Level surface of slope
2
1
Footing�has�u1 � 0,� u2 � � �, ur3 � 0
ur3 � 0
ur3 � 0Homogeneous clay slope
Homogeneous clay slope
Chapter 5
137Figure 5.3
Level surface of slope
u1 � 0
ur2 � 0
Slope surface of slope
Typical�meshing�of�finite�element�models�(for�the�cases�of��° � 45°, 60°, 75° and 90°)
Footing Level surface of slope
Homogeneous clay slope
u1 � 0
ur2 � 0
u1 � u2 � u3 � ur1 � ur2 � ur3 � 0
Footing�has�u1 � 0,� u2 � � �, ur3 � 0
ur3 � 0
ur3 � 0
Homogeneous clay slope
138Chapter 5
8−node plane strain quadrilateral, hybrid, linear pressure, reducedintegration element for strip footing problems (CPE8RH)
Figure 5.4 Element type used in ABAQUS modelling
By observing the load displacement response, a check was made to ensure that the ultimate
bearing capacity has been reached and that overall collapse has in fact occurred (i.e. the
load−displacement plot reaches a plateau, as shown in Figure 3.5).
5.4 ABAQUS FINITE ELEMENT ANALYSIS RESULTS
The computed finite element method results of the undrained bearing capacity of strip
footings on the surface of slope are presented in Table 5.1 and graphically in Figure 5.5 to
Figure 5.14. They express the bearing capacity in the form of the dimensionless parameter
qu��B depending on the ratio cu��B for each value of L�B (from 0 to 9) with smooth or
rough footings, where the slope angle �° varies from 15° to 90°.
The results indicate that, in general, the dimensionless bearing capacity qu��B increases
with (i) the increase of ratio cu��B, (ii) the increase of the ratio L�B, and (iii) the decrease
of the slope angle �°.
All these curves can be considered to have three main properties, namely (i) a linear section,
(ii) a non−linear section and (iii) a point where qu��B approaches zero.
In the linear portions the bearing capacity qu��B increases linearly with the ratio cu��B;
these regions are straight lines in Figure 5.5 to Figure 5.14. The linear portions of the
curves shown in Figure 5.5 to Figure 5.14 for cu��B � 0.75, 0.82, 1.35, 1.52, 1.93, and
139Chapter 5
2.11 for the cases of �° = 15°, 30°, 45°, 60°, 75°� and� 90°, respectively. In these sections,
the clay slope is strong enough (cu��B is larger than a certain value) so failure is by bearing
capacity failure and occurs either within the slope face or away from the top of the slope.
The failure mechanism is that of a general shear failure of a surface footing on a level
surface for larger L�B (Figure 5.17(b) and Figure 5.18(b)). For smaller values of L�B the
failure mechanism occurs within the slope face itself (Figure 5.15 and Figure 5.16).
Referring to Figure 5.15 to Figure 5.20, it can be seen that the slope height H is sufficiently
large for the bearing capacity of these strip footings to be independent of the slope height
H. The results can therefore be used for all geometrically similar slopes with H�B greater
than or equal the value used in the simulations which were 3 for the cases of
�=15° and 30°, and were 5 for the cases of �=45°, 60°, 75°� and� 90°. When the failure
is in the linear portion of the curve, the depth of the slope DH (Figure 5.1) does not affect
the bearing capacity.
In the linear section, for both rough and smooth footing cases, the ultimate bearing capacity
of the footings is governed by only local foundation failure (without overall stability of the
slope) even when the footings stand at the crest of the slope, L�B � 0 (Figure 5.15 to
Figure 5.16).
The relation between cu��B and qu��B can be approximated as a straight line function:
qu
�B� C1
cu
�B� C2 (5.3)
The values of the two coefficients C1 and C2 have been identified by a linear regression
to the ABAQUS data and are shown in the Table 5.2 for smooth footing cases and in the
Table 5.3 for the rough footing cases. The positive values of coefficient C1 indicate an
increase of bearing capacity qu��B with an increase of cu��B. For cases when L�B is large
enough, and the mode of failure is not affected by the slope, values of C1 are nearly the
same for all values of slope angle �°. In the linear section, generally when L�B is smaller
than 3, the larger the slope angle, the smaller the coefficient C1.
Chapter 5
140
� � 15° � � 30° � � 45° � � 60° � � 75° � � 90°
cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B
L/B=0
Smooth
10.000 46.959 10.000 41.557 10.000 35.989 10.000 30.606 10.000 25.247 10.000 19.9036.000 28.117 6.000 24.832 6.000 21.448 6.000 18.182 6.000 14.938 6.000 11.7292.000 9.286 2.000 8.093 2.000 6.884 2.000 5.725 2.000 4.602 2.000 3.5271.000 4.570 1.000 3.888 1.500 5.051 1.500 4.150 1.500 3.291 1.500 1.7940.700 2.953 0.700 2.400 1.000 2.517 1.200 3.121 1.300 2.272 1.390 1.3200.560 1.920 0.620 1.510 0.970 2.360 1.080 2.340 1.230 1.560 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=0
Rough
10.000 49.507 10.000 45.674 10.000 41.039 10.000 36.073 10.000 30.890 10.000 25.4096.000 29.667 6.000 27.314 6.000 24.447 6.000 21.392 6.000 18.229 6.000 14.9352.000 9.822 2.000 8.917 2.000 7.807 2.000 6.635 2.000 5.498 2.000 4.4001.000 4.853 1.000 4.265 1.500 5.693 1.500 4.740 1.500 3.859 1.500 2.6160.700 3.356 0.700 2.416 1.000 2.615 1.200 3.270 1.300 2.805 1.390 1.3200.560 1.920 0.620 1.510 0.970 2.360 1.080 2.340 1.230 1.560 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=1
Smooth
10.000 51.232 10.000 49.352 10.000 46.390 10.000 42.667 10.000 38.293 10.000 33.2826.000 30.731 6.000 29.559 6.000 27.696 6.000 25.345 6.000 22.560 6.000 19.3982.000 10.232 2.000 9.758 2.000 8.935 2.000 7.808 2.000 6.494 2.000 5.1561.000 5.103 1.000 4.734 1.500 6.504 1.500 5.032 1.500 3.690 1.500 1.7680.700 3.562 0.700 2.440 1.000 2.188 1.200 2.748 1.300 2.325 1.390 1.1200.560 1.920 0.620 1.510 0.970 2.060 1.080 1.740 1.230 1.560 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=1
Rough
10.000 51.956 10.000 51.080 10.000 48.478 10.000 45.202 10.000 40.974 10.000 35.9196.000 31.174 6.000 30.602 6.000 28.956 6.000 26.828 6.000 24.084 6.000 20.8712.000 10.382 2.000 10.121 2.000 9.333 2.000 8.117 2.000 6.695 2.000 5.3521.000 5.182 1.000 4.800 1.500 6.526 1.500 5.041 1.500 3.756 1.500 2.2840.700 3.620 0.700 2.471 1.000 2.559 1.200 3.200 1.300 2.325 1.390 1.3200.560 1.920 0.620 1.510 0.970 2.360 1.080 2.340 1.230 1.560 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
Table 5.1 Bearing capacity results for L/B=0 and 1
Chapter 5
141
� � 15° � � 30° � � 45° � � 60° � � 75° � � 90°
cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B
L/B=2
Smooth
10.000 51.966 10.000 52.210 10.000 52.071 10.000 49.389 10.000 45.593 10.000 40.8656.000 31.200 6.000 31.324 6.000 31.228 6.000 29.365 6.000 26.821 6.000 23.6782.000 10.402 2.000 10.443 2.000 10.140 2.000 8.523 2.000 6.764 2.000 5.1561.000 5.201 1.000 5.062 1.500 6.672 1.500 5.024 1.500 3.436 1.500 1.6390.700 3.641 0.700 2.579 1.000 1.987 1.200 2.546 1.300 1.888 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=2
Rough
10.000 52.440 10.000 52.991 10.000 52.393 10.000 50.367 10.000 47.097 10.000 42.5496.000 31.461 6.000 31.788 6.000 30.700 6.000 29.997 6.000 27.713 6.000 24.5732.000 10.489 2.000 10.606 2.000 10.254 2.000 8.594 2.000 6.843 2.000 5.1051.000 5.245 1.000 5.108 1.500 6.699 1.500 5.068 1.500 3.537 1.500 2.0300.700 3.671 0.700 2.610 1.000 2.092 1.200 2.637 1.300 2.081 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=3
Smooth
10.000 52.077 10.000 52.077 10.000 52.075 10.000 52.081 10.000 50.926 10.000 46.4096.000 31.206 6.000 31.247 6.000 31.244 6.000 31.248 6.000 29.925 6.000 26.7122.000 10.416 2.000 10.415 2.000 10.415 2.000 9.047 2.000 7.251 2.000 5.1751.000 5.208 1.000 5.208 1.500 6.998 1.500 5.305 1.500 3.631 1.500 1.6480.700 3.646 0.700 2.771 1.000 1.978 1.200 2.629 1.300 1.983 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=3
Rough
10.000 52.473 10.000 52.495 10.000 52.464 10.000 52.469 10.000 51.785 10.000 47.6586.000 31.471 6.000 31.503 6.000 31.500 6.000 31.504 6.000 30.059 6.000 27.3782.000 10.489 2.000 10.500 2.000 10.493 2.000 9.127 2.000 7.424 2.000 5.4101.000 5.250 1.000 5.250 1.500 7.031 1.500 5.378 1.500 3.857 1.500 2.0740.700 3.671 0.700 2.797 1.000 2.092 1.200 2.786 1.300 2.272 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
Table 5.1 (continued) Bearing capacity results L/B=2 and 3
Chapter 5
142
� � 15° � � 30° � � 45° � � 60° � � 75° � � 90°
cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B
L/B=4
Smooth
10.000 52.146 10.000 52.145 10.000 52.074 10.000 52.068 10.000 52.097 10.000 50.6636.000 31.287 6.000 31.288 6.000 31.450 6.000 31.242 6.000 31.258 6.000 29.0512.000 10.429 2.000 10.429 2.000 10.485 2.000 9.673 2.000 7.880 2.000 5.7901.000 5.215 1.000 5.215 1.500 7.418 1.500 5.783 1.500 4.036 1.500 1.9040.700 3.650 0.700 2.971 1.000 2.092 1.200 2.971 1.300 2.201 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=4
Rough
10.000 52.554 10.000 52.542 10.000 53.256 10.000 52.462 10.000 52.463 10.000 51.3556.000 31.532 6.000 31.539 6.000 31.942 6.000 31.478 6.000 31.493 6.000 29.5722.000 10.511 2.000 10.501 2.000 10.655 2.000 9.759 2.000 8.023 2.000 6.0821.000 5.256 1.000 5.257 1.500 7.478 1.500 5.868 1.500 4.249 1.500 2.4430.700 3.679 0.700 2.989 1.000 2.227 1.200 3.142 1.300 2.550 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=5
Smooth
10.000 52.125 10.000 52.125 10.000 52.070 10.000 52.116 10.000 52.112 10.000 52.1296.000 31.272 6.000 31.277 6.000 31.241 6.000 31.276 6.000 31.252 6.000 31.1782.000 10.422 2.000 10.425 2.000 10.414 2.000 10.356 2.000 8.687 2.000 6.6441.000 5.209 1.000 5.212 1.500 7.798 1.500 6.377 1.500 4.701 1.500 2.4790.700 3.648 0.700 3.156 1.000 2.221 1.200 3.467 1.300 2.779 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=5
Rough
10.000 52.546 10.000 52.541 10.000 52.461 10.000 52.554 10.000 52.510 10.000 52.5596.000 31.526 6.000 31.521 6.000 31.467 6.000 31.228 6.000 31.506 6.000 31.3082.000 10.509 2.000 10.504 2.000 10.489 2.000 10.396 2.000 8.830 2.000 6.9131.000 5.255 1.000 5.251 1.500 7.850 1.500 6.462 1.500 4.903 1.500 2.9990.700 3.678 0.700 3.170 1.000 2.333 1.200 3.623 1.300 3.108 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
Table 5.1 (continued) Bearing capacity results L/B=4 and 5
Chapter 5
143
� � 15° � � 30° � � 45° � � 60° � � 75° � � 90°
cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B
L/B=6
Smooth
10.000 52.121 10.000 52.108 10.000 52.075 10.000 52.073 10.000 52.069 10.000 52.0706.000 31.271 6.000 31.269 6.000 31.245 6.000 31.238 6.000 31.241 6.000 31.2432.000 10.424 2.000 10.423 2.000 10.415 2.000 10.414 2.000 9.504 2.000 7.6121.000 5.211 1.000 5.208 1.500 7.811 1.500 6.948 1.500 5.455 1.500 3.3630.700 3.648 0.700 3.327 1.000 2.344 1.200 3.977 1.300 3.497 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=6
Rough
10.000 52.510 10.000 52.537 10.000 52.484 10.000 52.477 10.000 52.456 10.000 52.4616.000 31.521 6.000 31.523 6.000 31.492 6.000 31.485 6.000 31.470 6.000 31.4772.000 10.508 2.000 10.507 2.000 10.497 2.000 10.496 2.000 9.631 2.000 7.8581.000 5.254 1.000 5.253 1.500 7.873 1.500 7.030 1.500 5.633 1.500 3.8110.700 3.676 0.700 3.339 1.000 2.442 1.200 4.110 1.300 3.768 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=7
Smooth
10.000 52.115 10.000 52.113 10.000 52.192 10.000 52.163 10.000 52.195 10.000 52.1936.000 31.270 6.000 31.269 6.000 31.316 6.000 31.316 6.000 31.315 6.000 31.3172.000 10.423 2.000 10.423 2.000 10.439 2.000 10.438 2.000 10.300 2.000 8.6451.000 5.211 1.000 5.212 1.500 7.829 1.500 7.544 1.500 6.210 1.500 4.3700.700 3.648 0.700 3.485 1.000 2.467 1.200 4.498 1.300 4.230 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=7
Rough
10.000 52.532 10.000 52.527 10.000 52.629 10.000 52.635 10.000 52.637 10.000 52.6206.000 31.520 6.000 31.517 6.000 31.582 6.000 31.576 6.000 31.580 6.000 31.5822.000 10.500 2.000 10.498 2.000 10.528 2.000 10.473 2.000 10.361 2.000 8.8701.000 5.257 1.000 5.254 1.500 7.890 1.500 7.603 1.500 6.371 1.500 4.7340.700 3.677 0.700 3.496 1.000 2.565 1.200 4.598 1.300 4.452 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
Table 5.1 (continued) Bearing capacity results L/B=6 and 7
Chapter 5
144
� � 15° � � 30° � � 45° � � 60° � � 75° � � 90°
cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B cu��B qu��B
L/B=8
Smooth
10.000 52.167 10.000 52.181 10.000 52.204 10.000 52.205 10.000 52.206 10.000 52.2026.000 31.280 6.000 31.280 6.000 31.340 6.000 31.323 6.000 31.324 6.000 31.3242.000 10.434 2.000 10.437 2.000 10.441 2.000 10.441 2.000 10.441 2.000 9.5621.000 5.211 1.000 5.210 1.500 7.831 1.500 7.831 1.500 6.914 1.500 5.2950.700 3.651 0.700 3.633 1.000 2.650 1.200 4.938 1.300 4.912 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=8
Rough
10.000 52.582 10.000 52.617 10.000 52.648 10.000 52.641 10.000 52.646 10.000 52.6406.000 31.511 6.000 31.555 6.000 31.587 6.000 31.585 6.000 31.564 6.000 31.6142.000 10.522 2.000 10.510 2.000 10.530 2.000 10.530 2.000 10.528 2.000 9.7251.000 5.261 1.000 5.262 1.500 7.898 1.500 7.897 1.500 7.042 1.500 5.5800.700 3.683 0.700 3.643 1.000 2.751 1.200 5.012 1.300 5.077 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=9
Smooth
10.000 52.138 10.000 52.137 10.000 52.266 10.000 52.269 10.000 52.231 10.000 52.2216.000 31.282 6.000 31.281 6.000 31.360 6.000 31.358 6.000 31.341 6.000 31.3352.000 10.428 2.000 10.427 2.000 10.453 2.000 10.452 2.000 10.447 2.000 10.4221.000 5.211 1.000 5.214 1.500 7.840 1.500 7.841 1.500 7.547 1.500 6.1860.700 3.650 0.700 3.649 1.000 3.034 1.200 4.925 1.300 5.504 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
L/B=9
Rough
10.000 52.567 10.000 52.523 10.000 52.767 10.000 52.773 10.000 52.736 10.000 52.7396.000 31.543 6.000 31.537 6.000 31.655 6.000 31.648 6.000 31.642 6.000 31.6452.000 10.515 2.000 10.512 2.000 10.555 2.000 10.555 2.000 10.542 2.000 10.4711.000 5.253 1.000 5.257 1.500 7.916 1.500 7.916 1.500 7.618 1.500 6.1860.700 3.680 0.700 3.680 1.000 2.938 1.200 5.384 1.300 5.627 1.390 1.0200.560 1.920 0.620 1.510 0.970 1.360 1.080 1.340 1.230 1.260 1.310 0.0000.490 0.000 0.540 0.000 0.925 0.000 0.980 0.000 1.120 0.000 1.310 0.000
Table 5.1 (continued) Bearing capacity results L/B=8 and 9
145Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
Figure 5.5 Bearing capacity of (a) smooth & (b) rough footings, (L/B=0)
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B=0
Figure 5.16(a)
Figure 5.16(b)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B=0
Figure 5.15(a)
Figure 5.15(b)
(a)
(b)
146Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B=1
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 1
Figure 5.6 Bearing capacity of (a) smooth & (b) rough footings, (L/B=1)
(a)
(b)
147Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B = 2
Figure 5.17(a)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 2
Figure 5.18(a)
Figure 5.7 Bearing capacity of (a) smooth & (b) rough footings, (L/B=2)
(a)
(b)
148Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B = 3
Figure 5.19(a)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 3
Figure 5.20(a)
(a)
(b)
Figure 5.8 Bearing capacity of (a) smooth & (b) rough footings, (L/B=3)
149Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B = 4
Figure 5.17(b)
(a)
(b)
Figure 5.9 Bearing capacity of (a) smooth & (b) rough footings, (L/B=4)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 4
Figure 5.18(b)
150Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B =5
(a)
(b)
Figure 5.10 Bearing capacity of (a) smooth & (b) rough footings, (L/B=5)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B =5
151Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B = 6
Figure 5.19(b)
(a)
(b)
Figure 5.11 Bearing capacity of (a) smooth & (b) rough footings, (L/B=6)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 6
Figure 5.20(b)
152Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B =7
(a)
(b)
Figure 5.12 Bearing capacity of (a) smooth & (b) rough footings, (L/B=7)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 7
153Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B = 8
(a)
(b)
Figure 5.13 Bearing capacity of (a) smooth & (b) rough footings, (L/B=8)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 8
154Chapter 5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Smooth footing, L/B =9
(a)
(b)
Figure 5.14 Bearing capacity of (a) smooth & (b) rough footings, (L/B=9)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 1 2 3 4 5 6 7 8 9 10
qu
�B
cu
�B
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Rough footing, L/B = 9
155Chapter 5
a)��° � 30°, cu��B � 6,� L�B � 0
Figure 5.15 Typical displacement vectors and contours for smooth footings
b)��° � 60°, cu��B � 6,� L�B � 0
156Chapter 5
a)��° � 30°, cu��B � 6,� L�B � 0
Figure 5.16 Typical displacement vectors and contours for rough footings
b)��° � 60°, cu��B � 6,� L�B � 0
157Chapter 5
a)��° � 45°, cu��B � 2,� L�B � 2
Figure 5.17 Typical displacement vectors and contours for smooth footingsb)��° � 45°, cu��B � 2,� L�B � 4
158Chapter 5
a)��° � 45°, cu��B � 2,� L�B � 2
Figure 5.18 Typical displacement vectors and contours for rough footings
b)��° � 45°, cu��B � 2,� L�B � 4
159Chapter 5
a)��° � 75°, cu��B � 6,� L�B � 3
Figure 5.19 Typical displacement vectors and contours for smooth footings
b)��° � 90°, cu��B � 6,� L�B � 6
160Chapter 5
a)��° � 75°, cu��B � 6,� L�B � 3
Figure 5.20 Typical displacement vectors and contours for rough footings
b)��° � 90°, cu��B � 6,� L�B � 6
Chapter 5
161
Slope angle � � 15° � � 30° � � 45° � � 60° � � 75° � � 90°
Linear section cu
�B� 0.75
cu
�B� 0.82 cu
�B� 1.35
cu
�B� 1.52
cu
�B� 1.93
cu
�B� 2.11
Coefficients C1 C2 C1 C2C1 C2 C1 C2 C1 C2 C1 C2
L/B=0 4.709 −0.132 4.183 −0.273 3.638 −0.393 3.110 −0.495 2.581 −0.559 2.047 −0.567
L/B=1 5.125 −0.019 4.949 −0.141 4.682 −0.429 4.357 −0.907 3.975 −1.455 3.516 −1.876
L/B=2 5.196 0.010 5.221 0.001 5.241 −0.3435 −5.108 −1.693 −4.854 −2.943 4.464 −3.772
L/B=3 5.208 0.000 5.208 −0.000 5.207 0.001 5.379 −1.712 5.459 −3.667 5.154 −5.133
L/B=4 5.215 0.000 5.215 0.000 5.199 0.088 5.299 −0.925 5.527 −3.174 5.609 −5.428
L/B=5 5.213 −0.003 5.213 0.000 5.207 0.000 5.220 −0.084 5.428 −2.169 5.686 −4.727
L/B=6 5.212 −0.001 5.211 0.002 5.208 −0.001 5.207 0.000 5.321 −1.137 5.557 −3.503
L/B=7 5.212 −0.001 5.211 0.001 5.219 0.001 5.216 0.006 5.237 −0.174 5.443 −2.241
L/B=8 5.217 0.001 5.218 0.001 5.220 0.000 5.220 0.001 5.221 0.000 5.330 −1.098
L/B=9 5.214 0.000 5.214 −0.001 5.227 0.000 5.227 −0.002 5.223 0.001 5.225 −0.028
Table 5.2 Bearing capacity coefficients C1 and C2 for smooth footings
Chapter 5
162
Slope angle� � 15° � � 30° � � 45° � � 60°
� � 75° � � 90°
Linear section cu
�B� 0.75
cu
�B� 0.82
cu
�B� 1.35
cu
�B� 1.52 cu
�B� 1.93
cu
�B� 2.11
Coefficients C1 C2 C1 C2 C1 C2 C1 C2 C1C2 C1
C2
L/B=0 4.961 −0.099 4.595 −0.273 4.154 −0.501 3.680 −0.725 3.174 −0.850 1.626 −0.852
L/B=1 5.197 −0.012 5.120 −0.119 4.893 −0.454 4.893 −1.154 4.285 −1.875 3.821 −2.290
L/B=2 5.244 0.001 5.298 0.010 5.267 −0.281 5.222 −1.849 5.032 −3.220 4.681 −4.256
L/B=3 5.248 −0.007 5.249 0.001 5.246 0.000 5.418 1.708 5.545 −3.666 5.281 −5.153
L/B=4 5.255 0.000 5.255 −0.010 5.325 0.005 5.338 −0.917 5.555 −3.086 5.659 −5.273
L/B=5 5.255 −0.001 5.255 −0.005 5.247 −0.004 5.270 −0.143 5.460 −2.090 5.706 −4.498
L/B=6 5.250 0.008 5.254 0.000 5.248 0.000 5.248 0.000 5.353 −1.075 5.575 −3.292
L/B=7 5.254 −0.008 5.254 −0.010 5.263 0.003 5.270 −0.067 5.285 −0.208 5.469 −2.067
L/B=8 5.258 0.007 5.263 −0.017 5.265 0.001 5.264 0.002 5.265 −0.001 5.364 −1.004
L/B=9 5.257 0.002 5.251 0.010 5.277 0.001 5.277 0.000 5.274 −0.007 5.284 −0.096
Table 5.3 Bearing capacity coefficients C1 and C2 for rough footings
163Chapter 5
The non−linear portions of the curves in Figure 5.5 to Figure 5.14 appear when cu��B �
0.75, 0.82, 1.35, 1.52, 1.93, and 2.11 for the cases of �° =
15°, 30°, 45°, 60°, 75°� and� 90°, respectively (Figure 5.21). The non−linear portions
shown graphically on plots in Figure 5.5 to Figure 5.14 reflects the complex interactions
between the footing bearing capacity and the overall slope stability and can be best
explained by referring to the magnified zone shown in Figure 5.21. The non−linear portion
of the curves in Figure 5.5 to Figure 5.14 occurs due to relatively week soil and the failure
types are mixed between local bearing capacity failure and global slope stability failure
depending on the distance L, the slope height H, and cu��B. The finite element method
results show that, in the non−linear sections, when cu��B is small, the type of failure tends
to be global slope failure and the bearing capacity is small. Conversely, when cu��B is
large, the type of failure tends to be local bearing capacity failure and the bearing capacity
of the footing is large. Note that in these cases, if the footing is absent from the slope, the
slope remains stable without any failure.
These transition points were identified by testing the domains with changes of the
parameters and watching results following these steps: (i) If the footing is absent from the
slope, the slope remains stable, this case belongs to the linear or non−linear sections. (ii)
When applying a certain value of displacement increments until the slope failure occurs.
If the model has global slope failure, it belongs to non−linear sections. Whereas, if the
model has local bearing capacity failure, it belongs to linear sections. (iii) change the value
cu��B until a point, at which the model sometimes has local bearing capacity failure and
sometimes has global slope failure, this point is transition points.
The non−linear sections end at points where bearing capacity is equal to zero (overall slope
failure).
The values of qu��B in Table 5.1 show that in the non−linear section the bearing capacity
qu��B is almost independent from position of footing on the slope L�B. This can be best
explained that in the non−linear portions, the model is governed by global slope failure.
164Chapter 5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
qu
�B
cu
�B
Figure 5.21 Bearing capacity of footing for various slope angles
�° � 15°
�° � 90°
�° � 30°
�° � 45°
�° � 60°
�° � 75°
Overall slope failure
Transition points
N HB
The overall slope failure points are the points when the curves (cu��B,qu��B) reach the
horizontal axis (i.e. qu��B =0), as shown in Figure 5.21. The finite element results show
that qu��B will be zero at cu��B= 0.49, 0.54, 0.925, 0.98, 1.12 and 1.31 for � =
15°, 30°, 45°, 60°, 75°� and� 90°, respectively. In order to determine these values, the
footing is removed from the slope, so gravity is the only force applied to the slope. When
cu��B is less than value of the overall slope failure point, overall slope failure occurs
(Figure 5.22). While cu��B is greater than value of the overall slope failure point, the slope
is stable and only fails at a certain load offer the footing is applied, qu��B >0.
According to the Shiau et al. (2008), the overall slope failure points have a relation to
stability number of Taylor (1937). Shiau et al. (2008) found:
N HB� cu
�HFs
HB� cu
�B for factor of safety Fs � 1 (5.4)
165Chapter 5
Figure 5.22 Typical displacement contours for overall failure without footings
c)��° � 45°, cu��B � (cu��B)qu��B�0
b)��° � 30°, cu��B � (cu��B)qu��B�0
a)��° � 15°, cu��B � (cu��B)qu��B�0
cu��B � 0.8
cu��B � 0.35
cu��B � 0.4
166Chapter 5
�° N H/BN H
B cu
�B (N HB� cu
�B)� cu
�B
15° 0.181 3 0.543 0.49 −10.816%
30° 0.181 3 0.543 0.54 −0.556%
45° 0.181 5 0.905 0.925 2.162%
60° 0.192 5 0.96 0.98 2.041%
75° 0.22 5 1.1 1.12 1.786%
90° 0.25 5 1.25 1.31 4.580%
Table 5.4 The relation between overall slope failure points and stability numbers
Table 5.4 shows the comparison values of overall slope failure points found from the finite
element method (using ABAQUS) with stability numbers N of Taylor chart (1937) for
HD � �.
The values of N HB
and cu
�B of finite element analysis in Table 5.4 are within 10.8% of each
other.
5.4.1 Effect of ratio L/B
In Figure 5.5 to Figure 5.10 when L�B � 5, the curves representing the relations
(cu��B,qu��B) for many cases of the slope angle �°, are far apart. When the ratio L�B
becomes larger, the curves converge together. For example, for a smooth footing, when:
L�B=2, cu��B =10, � � 15°, qu��B = 51.966 and L�B=2, cu��B =10, � � 90°, qu��B
= 40.865, while
L�B=5, cu��B =10, � � 15°, qu��B = 52.125 and L�B=2, cu��B =10, � � 90°, qu��B
= 52.129.
This can be explained by observing failure mechanism when the distance from the footing
to the crest of the slope L�B � 5, the failure occurs toward the slope and level surface
(Figure 5.15 and Figure 5.16) while for L�B � 5, failure occurs within the level surface
only, and hance the slope does not influence bearing capacity qu��B (i.e. bearing capacity
remains constant), (Figure 5.19(b) and Figure 5.20(b)).
However, for soft soil slope, (i.e. in the non−linear section) the failure zone depends on the
position of the footing L�B and the height of the slope H�B, but the bearing capacity
167Chapter 5
qu��B is nearly the same when L�B changed in each case of slope angle �° and strength
ratio cu��B (Table 5.1).
5.4.2 Effect of the footing roughness
In general, the bearing capacity of rough footings was found to be larger than that of smooth
footings (Figure 5.5 to Figure 5.14, and Table 5.1). However, these values are closer to
each other for the smaller slope angles �° and for larger distance ratios L�B. In the linear
portion, for otherwise identical problems with slope angle � � 45° and L�B � 5, the
bearing capacity qu��B of rough footings is higher than that of smooth footing by around
only 0.5 to 1%, and the domain of influence of rough footings is larger than of smooth
footings.
However, when the slope angle � � 45°, and the distance ratio L�B � 5, in the linear
section, the footing roughness contributes significantly to bearing capacity of footing. In
this region, qu��B for a perfect rough footing is larger than that for a smooth footing by
around 20% to 28%. This influence is reflected in the shape of the failure mechanism as
shown in Figure 5.15 and Figure 5.16.
In the non−linear section the roughness of the footing does not influence bearing capacity
very much. The reasons for this can be seen from many tests in non−linear section and
explained by the fact that the overall slope stability failure tends to occur rather than local
bearing capacity failure. Normally, in the linear section, instead of starting from footing
edge, the failure zone starts from a point in the level surface and ends at a point near the
base of the slope. After a certain increment of displacement, the overall footing failure
occurs and the bearing capacity approaches the same value with any type of footing/soil
interaction.
5.4.3 Effect of slope angle �°
In general, the bearing capacity qu��B decreases as the slope angle �° increases. The finite
element method results show that the bearing capacity of a strip footing on the level surface
of a slope is a strong function of the slope angle �° for L�B � 5. For distances L�B � 5,
in the linear section, the bearing capacity qu��B is larger if the slope angle �° is smaller.
However, for distances L�B � 5, in the linear section, bearing capacity qu��B is the same
even when the slope angle �° changes from 15° to 90°.
168Chapter 5
5.4.4 Comparison of this study with Narita and Yamaguchi
(1990) & Kusakabe (1981)
Narita and Yamaguchi (1990) investigated the bearing capacity of foundations on slopes
using log−spiral sliding surfaces as mentioned in the section 2.3.3 of Chapter 2. Results
from previous studies in this area are limited so cases of clay slope with � � 30° are
considered. ABAQUS results are compared with those of Narita and Yamaguchi (1990),
the simplified Bishop method, and the upper bound solutions of Kusakabe (1981). Note
that in these previous studies, the authors did not indicate what the footing/soil roughness
was. However, comparisons show that their studies are assumed to be for smooth footing.
cu��Bq��B
ABAQUS Narita &Yamaguchi
(1990)Log−spiral
Bishop Kusakabe (1981)Upper bound
25 104.30 107 104.4 102
5 20.64 21.1 20.8 20.2
1 3.91 3.94 4.03 3.84
0.5 0 − 1.92 1.78
Table 5.5 Comparison of F.E. results with analytical solutions (� � 0, L�B � 0,
� � 30°, L/B=0)
The results from finite element method (ABAQUS) presented in Table 5.5 are close to the
results of the log−spiral and Bishop methods. However, bearing capacity from the finite
element method solution is higher than that from upper bound solution of Kusakabe (1981)
by around 1%.
5.5 CONCLUSION
The bearing capacity of strip footings on clay slopes has been investigated using the finite
element method. The results obtained have been presented in terms of a ratio of bearing
capacity qu��B in both graphical and tabular form to facilitate their use in solving practical
design problems. The following conclusions can be made based on the finite element
results:
� In general, the dimensionless bearing capacity qu��B increases with the increase of
the ratio cu��B, the increase of L�B, and decrease of the slope angle �. A wide range
169Chapter 5
of these parameters has been investigated: the strength ratio cu��B ranges from 0 to
10, the ratio of distance from footing edge to the crest of the footing L�B ranges from
0 to 9 and the slope angle � varies from15° to 90° in increments of 15°.
� The curves describing the relationship between cu��B and qu��B can be characterised
using three main properties, namely a linear section, a non−linear section, and zero
bearing capacity point of the curves (overall slope failure).
� In the linear section, the bearing capacity qu��B is proportional to the strength ratio
of soil cu��B, and can be determined by Equation (5.3) with the two coefficents
C1 and C2, described in the Table 5.2 for smooth footings, and in the Table 5.3 for
rough footings. Within the linear section, local bearing capacity failure contained to
within the face of the slope. The failure occurs in the region above the toe of the slope;
the height of the slope does not influence the bearing capacity, meaning the bearing
capacity qu��B obtained from these simulations can be applied in other configurations
with larger heights. If L�B increases, the lines describing the relation (cu��B, qu��B)
are closer to each other, and when L�B � 5, the lines are nearly the same for all slope
angles. In the linear section, rough footings have higher bearing capacity than that of
smooth footings.
� In the non−linear section, the curves of the relation (cu��B, qu��B) stay far apart as
the failure mechanism is governed by overall slope failure. Slope angles do influence
the bearing capacity while footing/soil interation does not.
� Some of the results of this study compare well to the results of the previous
investigations by log−spiral, Bishop and upper bound method.
170Chapter 6
CHAPTER 6
CONCLUDING REMARKS
171Chapter 6
6.1 INTRODUCTION
Predicting the bearing capacity of foundations plays a central role in the design of many
civil engineering works. The current theoretical understanding of the bearing capacity of
the foundations is unsatisfactory in some respects. This work addresses this deficiency
through numerical simulation.
This work aims to provide a more rigorous set of numerical solutions to three common
bearing capacity problems: (i) the undrained bearing capacity of surface footings on
layered soils; (ii) the undrained bearing capacity of embedded footings; and (iii) the
undrained bearing capacity of footings near slopes. This work also aims to verify the
application of numerical software using finite element method in particular problems.
The majority of past research on bearing capacity of foundation has been experimentally
based and, as a result, current design practices are largely based on empiricism. Very few
rigorous numerical analyses have been performed to determine the ultimate bearing
capacity of foundations. This fact was clearly highlighted in the historical review in
Chapter 2. Furthermore, the majority of existing numerical studies are not considered
rigorous as they are based on approximate methods such as limiting equilibrium or the
method of characteristics (without extension of the stress field). A comparison of the results
obtained from previous studies provides an opportunity to validate the findings and
provides a truly comprehensive evaluation of the bearing capacity of a foundation.
6.2 UNDRAINED BEARING CAPACITY OF SURFACE
FOOTINGS ON LAYERED SOILS
The undrained bearing capacity of surface footings on layered soils has been presented in
the Chapter 3. The bearing capacity of strip, square and circular footings on two−layered
clays using the finite element method has been investigated. The results obtained have been
presented in terms of a modified bearing capacity factor N*c in both graphical and tabular
form to facilitate their use in solving practical design problems. The results showed that
two−layered clay problems can be effectively solved by the finite element method. In the
same system of H�B the ultimate bearing capacity increases as the relative ratio of cu1�cu2
decreases. In the same system of H�B and cu1�cu2, the ultimate bearing capacity of circular
footing is around 2% larger than that of a square footing, and that of strip footing is the
172Chapter 6
smallest. The ultimate bearing capacity of circular footings and square footings could not
be related to that of strip footing by a constant coefficient.
This work could be extended to include multi−layered soil, deep penetration, arbitrary
footing shapes, inclined footings, or finite footings spacing.
6.3 UNDRAINED BEARING CAPACITY OF
EMBEDDED FOOTINGS
The bearing capacity of embedded footings in clay has been investigated and is presented
in Chapter 4. The bearing capacity of embedded strip, circular, square and rectangular
footings in purely cohesive soil was studied. Results presented contribute to the
understanding of undrained bearing capacity of embedded footings.
The results obtained have been presented in terms of a normalised bearing capacity factors
Nc in both graphical and tabular form; and the shape factors Fcs; and the depth factors Fcd
to facilitate their use in solving practical design problems.
The range of footings parameters considered in this work covers most problems of interest.
However, only cohesive soil has been considered. In future work, cohesionless and
cohesion−frictional soil should be studied in a similar manner. The soil in this study is
homogeneous clay with constant cohesion. The finite element method provides an
opportunity to solve a more complex problems with increasing cohesion clay in depth or
layered soil.
6.4 UNDRAINED BEARING CAPACITY OF FOOTINGS
NEAR SLOPES
In Chapters 5 the undrained bearing capacity of strip footings on slopes with a wide range
of soil types and positions of footing on level surface was investigated.
To cover most problems of practical interest, solutions for strip footings on undrained soil
slopes with variety of slope angles were presented. For design purposes, parametric
equations have been provided that enable the bearing capacity of footings to be estimated
reliably. Such equations can be used to solve practical design problems.
The finite element package ABAQUS provides an ability to solve three−dimensional
problems of footing on level surface of the slope such as square, circular or rectangular
173Chapter 6
footing. The soil can be investigated not only for cohesive but also for cohesionless, and
cohesion−frictional soil. The depth of slope in this investigation is infinite, however, in
practical design, the depth can be limited and not a level base. The water table can be
modelled in the finite element problems to examine the effect of a position of the water table
to bearing capacity of footing on slope. These problems can be solved in future work by
the finite element method.
174Chapter 6
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